January 11, 20, February 3, and 10, 2022

*Long before the field of quantum information had progressed to the point of considering the materials science and engineering required to construct a physical quantum computer*—*a massively challenging task in its own right*—*researchers needed a theoretical basis to be convinced that such a computer could actually work. As much as anyone else, the theories advanced by Alexei Kitaev have built that foundation. Debates continue about how long it will take to achieve a scalable quantum computer, how we will know when we have achieved it, and even about what novel tasks quantum computers can complete. Wherever this endeavor goes from here, it is unmistakable that Kitaev's ideas helped launch the field.*

*Long before the field of quantum information had progressed to the point of considering the materials science and engineering required to construct a physical quantum computer*

*a massively challenging task in its own right*

*researchers needed a theoretical basis to be convinced that such a computer could actually work. As much as anyone else, the theories advanced by Alexei Kitaev have built that foundation. Debates continue about how long it will take to achieve a scalable quantum computer, how we will know when we have achieved it, and even about what novel tasks quantum computers can complete. Wherever this endeavor goes from here, it is unmistakable that Kitaev's ideas helped launch the field.*

*Kitaev grew up in the Soviet Union and he attended the Moscow Institute of Physics and Technology for college and the Landau Institute for Theoretical Physics, where he completed his thesis on quasicrystals in 1989. In the mid-late 1990s, when* *John Preskill* *was starting to think about quantum information theory, he met Kitaev at a conference, and they connected over anyons; Preskill worked on them in connection with his work in field theory, and Kitaev approached anyons in the context of fractional quantum Hall systems. Preskill invited Kitaev as a visitor to Caltech during his time as a researcher at Microsoft, and Kitaev joined the faculty in 2002.*

*Kitaev grew up in the Soviet Union and he attended the Moscow Institute of Physics and Technology for college and the Landau Institute for Theoretical Physics, where he completed his thesis on quasicrystals in 1989. In the mid-late 1990s, when*

*John Preskill*

*was starting to think about quantum information theory, he met Kitaev at a conference, and they connected over anyons; Preskill worked on them in connection with his work in field theory, and Kitaev approached anyons in the context of fractional quantum Hall systems. Preskill invited Kitaev as a visitor to Caltech during his time as a researcher at Microsoft, and Kitaev joined the faculty in 2002.*

*His research career at Caltech has grown in parallel with the Institute for Quantum Information and its expansion to the Institute for Quantum Information and Matter, and his collaborations with faculty at Caltech and with industrial partners such as Google have continuously pushed the field ever closer to creating quantum computers. Kitaev has also made major contributions in superconductivity and AdS/CFT theory, the latter of which has exciting implications for its ability to get us closer to a quantum theory of gravity.*

*His research career at Caltech has grown in parallel with the Institute for Quantum Information and its expansion to the Institute for Quantum Information and Matter, and his collaborations with faculty at Caltech and with industrial partners such as Google have continuously pushed the field ever closer to creating quantum computers. Kitaev has also made major contributions in superconductivity and AdS/CFT theory, the latter of which has exciting implications for its ability to get us closer to a quantum theory of gravity.*

*Kitaev's major recognitions include a MacArthur Fellowship in 2008, the Breakthrough Prize in 2012, the Dirac Medal in 2015, the Oliver E. Buckley Prize in Condensed Matter Physics in 2017, and his election to the National Academy of Sciences in 2021. Kitaev was part of the Breakthrough Prize's inaugural group of awardees, and his citation was "For the theoretical idea of implementing robust quantum memories and fault-tolerant quantum computation using topological quantum phases with anyons and unpaired Majorana modes."*

*Kitaev's major recognitions include a MacArthur Fellowship in 2008, the Breakthrough Prize in 2012, the Dirac Medal in 2015, the Oliver E. Buckley Prize in Condensed Matter Physics in 2017, and his election to the National Academy of Sciences in 2021. Kitaev was part of the Breakthrough Prize's inaugural group of awardees, and his citation was "For the theoretical idea of implementing robust quantum memories and fault-tolerant quantum computation using topological quantum phases with anyons and unpaired Majorana modes."*

### Interview Transcript

**DAVID ZIERLER:** This is David Zierler, Director of the Caltech Heritage Project. It is January 11, 2022. I am delighted to be here with Professor Alexei Kitaev. Alexei, it's great to be with you. Thank you for joining me.

**ALEXEI KITAEV:** Thank you for having me.

**ZIERLER:** To start, would you tell me your current title and affiliations here at Caltech?

**KITAEV:** I'm the Ronald and Maxine Linde Professor of Theoretical Physics and Mathematics. It's quite a long title. Had to remember. [Laugh]

**ZIERLER:** Of course, Ronald and Maxine Linde are wonderful, great supporters of Caltech. It must be such an honor to be named after them. Do you have an idea why you were named in honor of the Lindes? Is there any connection between your research and what they want to support at Caltech?

**KITAEV:** A few years ago, they said they wanted to support me, looking at my research and my biography. I don't know exactly why they decided to do so, but I'm very grateful that they did.

**ZIERLER:** As a professor in theoretical physics and mathematics, is that to say that in some regards, this is a joint appointment?

**KITAEV:** It's a courtesy appointment for mathematics. I'm a physicist, and I teach physics at Caltech.

**ZIERLER:** Why is it important for you to have a courtesy appointment in mathematics?

**KITAEV:** Maybe it's not that important, but I appreciate that the Caltech administration gave me this appointment. Previously, I held a joint appointment with computer science. I was doing some computer science, or rather, quantum computation, which was related to real computer science. But later, I focused entirely on physics, and that's why I changed to physics. But the fact that the PMA also gave me this courtesy appointment is very nice because I like mathematics, and I sometimes want to discuss mathematics with mathematicians. I'm not sure if I can make any impact on mathematics or the mathematical community here, but who knows?

**ZIERLER:** Just as a snapshot in time, January 2022, what are you currently working on?

**KITAEV:** I'm still working on the SYK model. This is a model invented a few years ago, around 2014, 2015, a many-body system, a quantum Hamiltonian. It's a pretty conventional system, not mysterious, but analogous, in some ways, to black holes, which are way more mysterious. Certainly, correlation functions for this model mimic those for a black hole. That's why I got interested. Of course, at that time, the hard part was to find a good model for that purpose. I have been studying it for several years, finding many interesting questions. But at this point, it might be that the interesting questions have been exhausted. I'm thinking about another problem, which is not very new, it's the classification of topological phases of matter. It's something that I worked on previously for several years, and it was initially connected to quantum computation and protected quantum memory. But eventually, it was pure theory work. There, there are some hard unresolved questions. I still hope I might be able to tackle them.

**ZIERLER:** This is as much a generational question as a scientific question, but in the way that, for example, John Preskill started in cosmology and came into quantum information, he had to develop part of what became quantum information. For you, as a student, was quantum information available to study?

**KITAEV:** No. I read the first article, actually an introduction to Manin's book, *Computable and Uncomputable* on mathematical logic. In the introduction, there were a couple paragraphs that contained a very exciting idea that was very novel at the time, that simulating quantum mechanics is difficult, and maybe one could use quantum mechanics to solve difficult computational problems. That idea was very exciting and caught my imagination. Unfortunately, I didn't know what to do with it. I was an undergraduate at the time. Later, I read Feynman's paper, and Feynman developed that idea somewhat, but eventually, I started doing quantum computation when the theory was created. The theory included the definition of a quantum computer as a quantum circuit. It also helped that in grad school, I learned some computational complexity theory. That was a good background to start working on quantum computation.

**ZIERLER:** Does the work in SYK theory get you a little closer to the world of observation and experimentation than you might have been earlier in your career?

**KITAEV:** It was closer to what I studied previously. My main background is condensed matter. I studied in Gor'kov's theory group at the Landau Institute. The main focus there was condensed matter. The SYK model, in some sense, belongs to condensed-matter physics. Of course, it's a bit abstract for a usual condensed matter system. But I used the same methods, plus some field theory methods I learned. But I don't know much of field theory.

**ZIERLER:** At the institutional level, what's going on with IQIM right now? What are some of the big projects its members are involved in?

**KITAEV:** I cannot speak about the whole project, John Preskill would give you a better answer. But I know something that people at IQIM did, in particular, in pure theory, which is quite interesting. One work that included John was holographic codes, which might also be related to quantum gravity. They're even more abstract than the SYK model. [Laugh] But it's pretty interesting. Thomas Vidick has worked on complexity theory related to quantum computation. There's some experimental work that's not related exactly to quantum computation, it's more condensed-matter physics. David Hsieh is doing experiments using laser pulses. But I don't really know much, and I don't want to give incorrect information about what other people are doing.

### Error Correction Between Here and Quantum Computers

**ZIERLER:** So many questions about your sense of the state of quantum computation today. There are many, many opinions. First, at the most basic level, do we have quantum computers yet the way you define them?

**KITAEV:** No, we don't. Current quantum computers lack error correction. They don't have this error correction machinery that is necessary to scale them up. People are working on this problem. Actually, I work part-time at Google Quantum AI in Santa Barbara, and they're building a quantum computer using superconducting qubits. The state of the art is about 50 qubits, but those qubits are not error-corrected. More recently, they built a quantum error-correcting circuit, quantum error-correcting code. It's a variant of a surface code, something I worked on theoretically a number of years ago. Now, it's realized in practice. The first version is not perfect, but hopefully, progress will be made. But it's a long road. I don't expect a practical computer that could solve real problems will be built soon.

**ZIERLER:** On this long road, what is your sense of the work that needs to be done, both from an engineering basis and a theoretical basis before true quantum computation is achieved?

**KITAEV:** I actually don't know what to expect because there are different options. Right now, it looks like there isn't a good technology to build a quantum computer. There are some technologies that can be used for that purpose. From my perspective, the situation is similar to trying to build a classical computer in the 19th century. People tried that, Charles Babbage, in particular, who had the idea of a universal classical computer in the middle of the 19th century. He didn't have enough resources, or he might've succeeded. I don't know. But it would have been very awkward because the computer would've used mechanical gears. Currently, it looks like the quantum computers people are building will look like that, some big, very inefficient machine. Of course, if it works, it will be great. But it will be big and expensive. It would be much better if a new technology emerged. It might still emerge in the following years. Within ten years, the perspective may change completely, we'll have this new technology, and things will be much easier.

**ZIERLER:** You mentioned it will be great once quantum computers are able to solve real problems. Do we have an understanding of what kinds of problems quantum computers will be good at solving or at least better than classical computers?

**KITAEV:** Yeah. One class of problems is actually bad. It's breaking codes, like RSA. This is a code that's used to encrypt commercial transactions. If it's broken, it's not a good application. But there are also good applications. Simulating quantum mechanical systems, wherever it's useful. One possible application is predicting properties of molecules without actually synthesizing them. One would predict what color a substance will be as well as other properties. That, as well as predicting properties of materials or nanoscale devices, can be very useful. Actually, many devices involve quantum mechanics in terms of materials properties. But usually, once you have measured those properties, you can proceed with classical mechanics to understand how the device works. But when the devices become very small, quantum mechanics becomes more important. One can possibly use a quantum computer to simulate that. As a physicist, I look forward to simulating systems where experiments are hard. Like in the high-energy domain, or maybe one could even simulate string theory, although I don't know if that would be possible with a quantum computer.

**ZIERLER:** That would be very exciting.

**KITAEV:** Yeah, that would be exciting. But this is just speculation. More realistically, one could simulate field theories, properties of the universe in early stages, when the temperature was very high. Simulating things like high-Tc superconductors. People tried to do that experimentally by making sophisticated measurements and using classical computers. High-temperature superconductors are so-called cuprate materials that include planes of copper and oxygen. They have been studied for more than 30 years, and they're still puzzling. It's not understood how they work. There are many theories, and there are some conflicts between different theories and between those theories and the experiment. Making accurate simulations of the materials could help clarify the picture.

**ZIERLER:** As you well know, Google, Honeywell, Amazon, IBM… there are major companies pouring significant resources into developing quantum computers. Is your sense that that this is more motivated from a basic science approach, the way that Bell Labs funded research that had value later on, or are these companies already articulating how they might monetize quantum computation?

**KITAEV:** I'm not sure. I think there are several elements here. One important element is to be ready for the next revolution. They have expertise in quantum computers. As they became practical, they can monetize them. It won't happen immediately. Google, I believe, is investing significant resources to actually build a quantum computer from scratch. It's not just waiting until the technology emerges, they're actually developing the technology. But for some companies, it might be different. There's also a big element of hype.

**ZIERLER:** As you mentioned, there are many exciting possibilities so far about what quantum computers can do for simulating some of the most intractable problems in physics, high-Tc, quantum gravity, string theory. But from a philosophical perspective, how much progress will that actually allow us? In other words, a simulation is still not yet an experiment that can be viewed and understood, and data to be parsed out by researchers. Or is it? Is there something different about a simulation that comes from a quantum computer? When we simulate something, is that the same or different than an experiment that bears out a theory?

**KITAEV:** In general, it's different. In general, we might not know exactly how nature works, and we might simulate something, but it would be not the actual thing. In many cases, we're confident that we will be simulating the right thing. The laws of physics, apart from high energy, are well-understood. To simulate something with atoms, we know enough of fundamental physics to have a complete system of equations. We have to make some approximations. Simulating every atom in a system is impractical. By making those approximations, we might miss something important. In principle, with quantum simulation, we can be more confident than with classical simulation because when putting the system on a classical computer, we have to do huge approximations, and we necessarily lose details. We cannot be sure whether these details are important or not. On a quantum computer, it will be a more faithful simulation.

**ZIERLER:** String theorists, of course, will tell us we must be patient, that they're operating at energies or in dimensions that are not currently possible with the tools that are available to us. Will the simulations that might be achievable with a quantum computer satisfy those people who say string theory has never shown us any actual connection to nature, or will that argument continue, and quantum computers will not satisfy those people?

**KITAEV:** One can only imagine that in the future, people will just use string theory and field theory simulated on a quantum computer and reproduce all fundamental laws of physics. They will be able to calculate the fine-structure constant and other parameters of the universe. But I don't believe that will happen. I think string theory is just an abstract theory. Even string theorists think there are many parameters that need to be specified to describe the universe because there's a so-called energy landscape. In a single fundamental theory, there are many solutions. Many possible universes, so to speak. We live in a particular universe. To know which universe we live in, we need to know an amount of information that is very hard to get. I don't think just simulation will solve this problem. Besides this problem, in string theory, it's not known whether string theory is indeed the fundamental theory of nature. It might be just a nice mathematical theory, but that nature might work in a different way.

**ZIERLER:** If you had to guess long into the future, do you see quantum computers as playing a role if we are ever to understand how to integrate quantum mechanics and general relativity?

**KITAEV:** I don't think quantum computers will solve this problem. It's a theoretical physics problem. In some ways, quantum mechanics and general relativity or gravity are integrated because one can quantize gravitational fields like quantizing gravitational waves and getting gravitons. Problems arise when we try to describe gravitational physics at very high energies, close to the Planck energy. Things become inconsistent. We try to combine these two theories naively. String theory is an attempt to combine them in a nontrivial way such that they remain consistent. But I don't think that quantum computers can help this. Maybe at some point, when there are several competing theories, simulations could help choose the more plausible theory.

**ZIERLER:** A question more closely related to your area of expertise. What is topological quantum computation? In other words, what was quantum computation before you developed a concept of topological quantum computation?

**KITAEV:** I would say quantum computation is a process where we have a quantum state encoded in some way, and we act by unitary operations on that quantum state. We can also do measurements in the middle of computation or postpone them until the very end. Topological quantum computation is a way to encode quantum information and work with the encoded information such that it remains protected. It's a variant of error correction or fault-tolerant quantum computation. It's necessary both for classical and quantum computation to protect information from errors. In classical systems, information can be protected from errors pretty easily. It's such a simple thing that we may even forget about it because when we think about memory cells, a bit, 0 or 1, is represented by hundreds of electrons. If it's 100 or 101 electrons, it doesn't matter. This gives a high degree of robustness to classical memory cells. With quantum computers, one needs to invent other ways to protect qubits. Representing a qubit by a large system doesn't work. It helps us to avoid one type of error, but it engenders another type of error, the so called phase error. People have invented quantum error-correcting codes to protect quantum information. My invention was topological quantum computation, in particular, quantum computation using anyons and Majorana fermions. That is a quantum error-correcting code implemented by a physical system.

The physical system itself has some error-correcting features. It's related to topology because, in particular, computation with anyons works this way. We have anyons, particles in a two-dimensional system. It's an imaginary picture, nobody has realized it yet. But we move those particles around each other in different processes. One process, where two particles stay put or wiggle a little bit, is topologically different from the process where the first particle circles around the second particle. This circling can also be described as braiding if we draw a three-dimensional picture. But in the end, it's a type of error correction, or quantum encoding. This encoding has some topological features. Topology refers to different types of curves, different paths. One path goes in circles, and the other doesn't.

**ZIERLER:** When we're talking about quantum computation as protecting quantum information from errors, it almost sounds like this is a defensive mechanism, where errors are otherwise attacking the quantum information. I wonder if that's a metaphor that makes sense here.

**KITAEV:** From the practical perspective, it's certainly a good metaphor. We need to fight errors.

### The Source of Errors in Quantum Information

**ZIERLER:** Where do the errors come from? And how do we even know they're errors? Errors according to what?

**KITAEV:** That's a good question. Of course, to discuss errors, one needs to focus on a particular physical implementation. There are many implementations using atoms and superconductors. Let me describe one implementation that uses quantum dots. It's so sensitive to errors that it will hardly work in practice. But one can imagine a device where an electron can be in two potential wells or two places. Say, in the plane or even on a line. An electron can move along a line, and there are some gates or pieces of metal to which some voltage is applied, and they confine the electron such that it can be either between wires one and two or wires two and three. It's hard for the electron to move from one place to the other place. Classically, it seems like a good protection from an error. In this case, the error is a spontaneous jump of the electron from one place to the other. It's indeed a possible error, and in classical memory, one also has to do this kind of protection. In flash memory or in CCDs, electrons are trapped at certain places in the semiconductor structure. It must be hard for them to move to a different place. But quantum mechanically, it's not enough to prevent them from moving because there is a different type of error called phase error. The phase error occurs when other particles around interact with the electron. This depends on the electron's position. A good metaphor is that the environment is watching the electron. As soon as it learns where the electron is, the quantum information is destroyed. It's only safe when nobody's watching.

**ZIERLER:** What does that mean, when nobody's watching?

**KITAEV:** There should be no way for this information to leak to the environment or anywhere. It should remain confined to some well-controlled system. When we actually want to measure it, it's fine to remove this protection and measure the system. But it's very difficult to protect an electron in a quantum dot from interacting with the environment. I could try to describe quantum error-correcting codes, but I guess it's not very appropriate here.

**ZIERLER:** No, please, it would be great.

**KITAEV:** I'm teaching error correction right now, and the simplest way to protect a quantum state from errors is based on repetition and dual-repetition codes combined, and that is called Shor's code. Peter Shor invented this error-creating code in 1995. The idea is this. It would talk about zeroes and ones. First, we build protection from bit-flips, when zero becomes one spontaneously, by repeating the same bit several times. You will write zero, zero, zero, and there is a chance that one of those zeroes turns into one, but it's OK because it's only one of the bits that will be in error. We can look at the three bits and see that we have two zeroes and one one. That one must be wrong, and the other two are right; that's how classical error correction works. But in the quantum world, the phase error destroys information because the environment looks at the string zero, zero, zero, or one, one, one, and it can just look at the first bit and tell whether it's zero or one. But we can use a duality between zeroes and ones and quantum superpositions called plus and minus.

The plus is a superposition of zero and one with positive coefficients, and minus is a superposition with one of the coefficients negative. The phase error may be described as changing a coefficient in the superposition. The coefficient of zero doesn't change but the coefficient of one changes sign. If the environment looks at the superposition of zero and one with the plus sign, it changes the sign of one from plus to minus, and it becomes the negative superposition. The phase error flips the qubit between the plus and minus states. It doesn't flip between zero and one, but it changes the plus to the minus and the minus to the plus. We can use the same idea of repetition, but this time, we repeat plus, plus, plus, or minus, minus, minus. One can have two layers of protection. One layer protects from bit-flips, and the other layer protects from phase errors. Each layer requires that a bit is repeated three times. The combination of two layers requires that the bit is repeated nine times. It's a nine-qubit code. This is one type of code.

A more practical code is called a surface code, where physical qubits, in which information is encoded, are put as an array in a square, and some interaction between them is arranged such that those qubits are correlated. In my previous example, the correlation was: if one is zero, then the next should also be zero. If one is plus, the next should also be plus. But there, the correlation is more complex. They're correlated in such a way that a single error doesn't do much to the encoded information. Or rather, it leaves traces that can be seen, and the error can be corrected on the spot. The only bad thing that can happen, if an error affects a whole row or column of qubits. Then, it leaves no trace, and we won't be able to correct it.

**ZIERLER:** The idea that quantum information can be protected, what does that look like? What's the value of having protected quantum information?

**KITAEV:** It's a necessity because if we want to do quantum computation, we want to store quantum information for some time, at least as long as the computation takes. It may take, say, a few seconds. Even that is considered a long time. And a logical zero or a logical plus, a combination of a zero and a one, should remain the same from the beginning to the point when this information is used. And to make sure it doesn't change, it remains the same, we must protect it from errors. If there are errors, this zero or plus becomes a random state.

**ZIERLER:** The idea that quantum information can be destroyed, is this related at all to the information that goes into a black hole? Is that information also destroyed, or is it simply understood as unrecoverable?

**KITAEV:** Unrecoverable in the practical sense. Basically, our qubit interacts with the environment, or some parts of our computer, and they become entangled in the sense of quantum entanglement. Those parts become entangled with other parts and so on. It quickly turns into such a tangle that nobody will be able to undo it.

**ZIERLER:** You mentioned two terms that were of particular value for developing topological quantum computing. Let's start, first, with anyons. What are anyons?

**KITAEV:** Anyons are quasi-particles that occur in two-dimensional systems. Let me define quasi-particles in general. We have intuition for particles. Particles are objects moving in free space. But we can also think what free space is. That is not a trivial question. Because something can propagate, not as particles but as a kind of object, in a medium. Waves propagate on the surface of water, sound waves propagate in air. When a wave propagates, nothing physically moves from the source to the destination. It's just vibrations of molecules of the air, and those vibrations propagate. But individual molecules don't move far. They move a little bit. We have these two pictures. Propagating waves, where there's a medium that vibrates, but nothing moves end-to-end, and the picture where a particle moves. In quantum mechanics, and in particular, when we consider many-body systems or relativistic systems, these two pictures are just two ways to describe the same process. In fact, a vacuum is a kind of medium where things propagate, and more intuitively, crystals are a medium where electrons can propagate. But one can imagine that the thing that propagates is more similar to a wave rather than a particle, although in quantum mechanics, there's no difference, it's all the same.

**ZIERLER:** Why are they all the same in quantum mechanics?

**KITAEV:** It's a bit hard to explain, but let me give a concrete example of a quasi-particle. Let's begin with a seemingly trivial case where in a crystal, say, in an insulator like silicon or diamond, one puts an extra electron in, and it moves. But the crystal already contains a lot of electrons that are indistinguishable from the new one, so we cannot tell whether it's a single runner or a relay. We can also create a hole by removing one of these electrons, and that hole, in the regular pattern of electrons, will behave like a particle. For example, the hole moving from atom one to atom two means an electron jumping from atom two to atom one. The space it leaves on atom two is the new hole, and the old one is filled. Thus, a hole effectively moves from one place to another. And this is an example of a quasi-particle. Quasi-particles are changes in the electron pattern. Those patterns and changes can be very different and quite complex. Coming back to your original question about anyons, anyons are special quasi-particles that exist in certain systems, like under the condition of the fractional quantum Hall effect. This is one example. Quantum Hall systems are two-dimensional systems where electrons move in a plane, and one popular material is graphene.

It's a sheet of carbon one atom thick. One can change the number or density of electrons. Depending on the density and the applied magnetic fields, different electron patterns form. In certain conditions, there are quasi-particles with fractional charge which are already peculiar. The charge can be, say, one-third of the electron charge. It means that the pattern of electrons in the material changes such that around one point, it's slightly more crowded. It's not exactly at the point, it's kind of spread out in a small region. But this blob or partially localized electric charge can move thorough the system. Fractional charge is quite peculiar. But the most peculiar property is this, which defines anyons. If one particle moves around another particle and makes a full turn, then a nontrivial change of the quantum state occurs as if the circling particle senses the presence of the other particle in the middle. This change can be pretty simple, like the wave function of the whole system picks up a phase factor. But it can be more complex, and when it is, those anyons are called non-Abelian. With non-Abelian anyons, even when there are two particles at fixed positions, this knowledge does not specify the quantum state completely.

The pattern of electrons between those quasi-particles is not completely determined by the positions of the quasi-particles. There are some hidden degrees of freedom. When we move one particle around the other, it's like we turn a crank, and we change some internal degrees of freedom in the electron pattern. That can be a nontrivial change, not just a phase factor of the wave function. It can be an actual change of the wave function. The nice thing is, these degrees of freedom are hidden from measurement, but they're also hidden from unwanted interactions. They're immune to errors. If we can control the quasi-particles, we control those hidden degrees of freedom implicitly. We can control them by moving quasi-particles around each other. That is topological quantum computation.

**ZIERLER:** You mentioned Majorana fermions. Are these the same fermions we know from particle physics?

**KITAEV:** There's a mathematical relation. They're not exactly the same things. Majorana fermions that are used for quantum error correction are better called Majorana modes. These are places where some change can occur, or access points to the hidden state. However, there should be two modes involved to make such a change. If those modes are separated by a sufficient distance, it's unlikely that the change will occur by itself. We need to arrange for it to occur. That explains the protection from errors.

**ZIERLER:** Because these concepts are so abstract, how do you measure progress in the field? How do you know if you're on the right track? What are the parameters that you use?

**KITAEV:** I should say these things are abstract, but this is a measure of effort you need to take to learn those things. Once you've learned them, they're not completely simple, but they're not so complicated either that you can't understand them. You can clearly see what works and what doesn't work, in most cases. Theoretical progress is measured by understanding various things and by developing new concepts, testing them, and solving various problems. But of course, you also need to measure the experimental progress.

**ZIERLER:** For problems that are not very abstract, that are very concrete, for example, the holy grail of achieving high-Tc superconductivity, what would you be looking for in quantum computers to be able to make that achievement?

**KITAEV:** I'm not an expert in high-Tc superconductors. My understanding is that the problem is hard because first, there is no simple theoretical solution. For ordinary superconductors, so-called Bardeen-Cooper-Schrieffer superconductors, there is a simple theory invented by Bardeen, Cooper, and Schrieffer, and it describes the ground state, and it describes excitations. It's a pretty simple mathematical theory. For cuprates, planes of copper and oxygen that are responsible for the superconductivity, there's no such simple model and mathematical description of the quantum state. Since there is no model, it's also not clear what is important and what is not. There are various features. The planes of copper and oxygen are not the only things that are present in the material. There are other atoms. Those atoms are not arranged in a regular pattern, it's actually a disordered system.

One can ask whether the disorder is important. There are many factors that might not be important. Experimentalists have tried to determine which factors are important and which are not and have made a lot of progress over 35 years or so, but they have not completely resolved the puzzle because there are many relevant factors. In a quantum computer, it may be possible to make a simple system, as simple as one can imagine, this lattice of copper and oxygen, and each atom may be described by just a site, which may be occupied with one or two electrons, and those electrons can jump. One can make a simple model, run a simulation, and see whether it has the desired property of a superconductor. And this simplification would help make a lot of progress. Then, one would try to dissect the problem even further, change some parameters in the model to see whether those parameters are important or not. In a real material, it's much more difficult to make such changes.

**ZIERLER:** Another real-world challenge, as you well know, data storage and classical computing, the cloud, as we say, is extraordinarily energy-intensive. Do you envision a future where quantum computers significantly decrease the energy requirements of data storage?

**KITAEV:** No, I don't expect that. The servers that are used right now do not require quantum computing. We just need to develop other technologies that would make memory slightly slower but more energy efficient. And that is happening, I suppose.

**ZIERLER:** Do you think, though, that quantum computers will replace classical computers in the way that we store our data? Or are those really two separate worlds?

**KITAEV:** No, quantum computers will only be used for programs where classical computers are inefficient. Most practical problems we solve right now can be solved on a classical computer efficiently. There are problems we would like to solve that neither classical nor quantum computers can do well, such as NP-complete problems, for example, or many optimization problems. It's possible to optimize a function of many variables to some degree, but if you want to find the absolute minimum or maximum, it requires exponential computation. It's not known whether a quantum computer can help with such problems. There's a small niche for quantum computers where they can do much better than classical computers. I described some of those problems: code-breaking and simulating quantum mechanical systems. Unfortunately, we don't know very many such problems. These two problems I mentioned are the main ones known at the time.

**ZIERLER:** We talked about quantizing gravity and the effort to establish, perhaps, a grand unified theory of physics. Is there a grand unified theory for quantum computers? In other words, do the theories that you've developed have competing theories?

**KITAEV:** No, the basic theory was developed in the 1990s and still holds. Actually, it started in the 1980s, when the circuit model was proposed. The circuit model was based on ideas of Manin and Feynman because Manin's work was not well-known. He didn't develop this idea as much. But Feynman's ideas were formalized by David Deutsch in the 1980s, and that created the foundation for the theory of quantum computing. It's a model of computing, and there was some work on different models and equivalencies between models. We have a solid foundation in the sense that even though one can propose a slightly different model, it can be shown to be equivalent to the previous one. All good models are equivalent to each other for quantum computing. Then, some examples of quantum algorithms were found, in particular, Peter Shor found an algorithm to break an integer number into prime factors. That can be used for breaking codes. There are other examples of quantum algorithms. As a third component of the theory, we know how to deal with errors. There's a theory of quantum error-correcting codes and fault-tolerant quantum computing. Fault-tolerant means we compute and correct errors simultaneously. Those errors happen during computation and even during the error correction, but we can keep up with errors and correct them faster than they come. That's fault-tolerant quantum computation. All this theory was created in the 1990s, and some improvements were done later. So, the foundations have been established and are unlikely to change.

**ZIERLER:** We've all seen photographs, for example, of Google's quantum computer, which looks like something out of a science fiction movie or something that an alien civilization would put together. Is your sense that, ultimately, quantum computation is going to be something that's usable by an individual? Will there be a quantum computer laptop or phone, for example?

**KITAEV:** I don't know. I don't see viable applications for a quantum computer in a phone. Not to mention that it's very expensive. [Laugh] Right now, it seems there are some niche applications for a quantum computer. But one application is very important to me as a physicist, namely, to simulate physical systems that are hard to study otherwise. That's why I'm working on building a quantum computer. Well, right now, I'm not actively working on it as a Caltech professor. I do some applied work as part of Google. But as a physicist, my main interest is in studying physical models of various things.

**ZIERLER:** To think back 20, 25 years ago, the state of IQI, what were some of the real low-hanging fruit, the really fundamental things to work on in quantum information, and how have those things developed over time?

**KITAEV:** You're talking about quantum information in general, not restricted to quantum computing? If we go back to 1995, at that time, not all fundamental problems had been solved. It was an exciting time. Not even the theory of quantum error correction had been developed. On this front, there have been great developments in terms of error-correcting codes, how to use error-correcting codes. John Preskill, Daniel Gottesman, and other people including myself participated in that. We wrote some papers about the use of surface codes, anyons, topological computation. That is something that I have worked on, and it's related to computation. Among other things, I particularly like is the work of Preskill and Hayden that makes a connection between quantum information and black holes. This is a really exciting paper. They described how to recover information, in principle, from a black hole, or any complex system where information seems to be lost.

If we have a reference system that is operated in the same way, driven by the same Hamiltonian, even being a quantum computer simulation, and is entangled with the physical system, then under various assumptions, we can recover information that appears to be lost in the physical system. Preskill and Hayden put it in a very nice way, they said that Alice wants to destroy a diary containing her private notes, and Bob is trying to spy on Alice and recover the information that seems to be destroyed. Alice thinks that the best way to actually destroy the information is to throw the diary into a black hole. But it turns out that since black holes emit particles called Hawking radiation, one can, in principle, recover the hidden information using those particles. Hawking radiation alone does not allow us to recover the information easily, but if there's another object, say a quantum computer as big as the black hole itself, and its qubits are perfectly entangled with the black hole, then the recovery is possible. Well, the black hole is not exactly made of qubits, but it's still a quantum state.

Suppose we have an entangled quantum state, and we know how the black hole works, and Bob knows, then Bob can use this prior arrangement together with a few photons emitted by the black hole and captured, and this will allow him to reconstruct the new state of the quantum black hole. The new state is the state that includes Alice's diary. This is an amazing paper, and it had a lot of follow-up results. Also, there was some work in quantum optics. Jeff Kimble has done a lot of experimental work in his lab, and he was part of IQI and IQIM for a long time. The greatest impact, I think, is that there were many post-docs who worked at IQIM and previously IQI, and now they've become renowned physicists and experts on quantum information. I think Caltech is the biggest school that taught quantum information physicists in the world. I don't have quantitative measures, but I have this feeling that we have the strongest group of people. [Laugh]

**ZIERLER:** The complex systems you're referring to with Preskill and Hayden's work, are those the same as the many-body systems with the topological properties you referred to earlier? Are many-body systems and complex systems interchangeable?

**KITAEV:** I think when people say "many-body systems," they don't necessarily mean the same thing as "complex systems" in the following sense. A many-body system can be described by a simple model such as a periodic array of atoms. To talk about the tradition of how people use these words, many-body physics previously meant classical many-body physics of, say, several planets that interact with each other gravitationally. Later, quantum many-body physics was developed. In quantum many-body physics, there are multiple atoms or electrons, but one can study things mathematically and understand them to great detail using simple models. When people say "complex systems", to me, it sounds like something messy, too complex to understand well. I don't know. That's why I prefer to say "many-body systems."

### Real and Toy Black Holes

**ZIERLER:** Earlier in the 20th century, black holes were theoretical or mathematical concepts. But now, we have observational evidence that they exist. Have there been recent advances, like the imaging by the Event Horizon Telescope or the amazing work done by LIGO, that have been relevant for the kinds of questions you're asking about black holes?

**KITAEV:** No, these are different questions. The questions I'm asking, or that work by Hayden and Preskill, are related to something very hard to observe experimentally. Basically, we're pushing this interplay between quantum mechanics and gravity to the limit, seeing how it works and whether it's consistent. So far, it has been consistent, but we've only explored certain aspects of this interplay. The black holes that are observed in our universe are big, quantum effects in them are very hard to measure. These are different things, unfortunately.

**ZIERLER:** A real history of physics question as you work towards quantizing gravity and utilizing black holes. How different fundamentally are the questions you're asking from those that were raised by Albert Einstein and Niels Bohr so many years ago? Are they fundamentally the same questions, or have there been advances in the field where they're different?

**KITAEV:** Those original questions have long been solved. Of course, we're thinking about new questions, otherwise there would be no progress in physics. [Laugh]

**ZIERLER:** Very briefly, what's been solved, and what now can be asked as a result of those questions being solved?

**KITAEV:** Since Heisenberg, Schrödinger, other great physicists in the beginning of the 20th century, we know what quantum mechanics is. You are referring to Einstein and Bohr, and they were also thinking about quantum mechanics before it was formulated as a complete mathematical theory. But in the late 1920s, quantum mechanics had been established. In the decades later, it was applied to many physics problems. Right now, there are some unresolved questions, and the most important question is to understand how gravity works at the microscopic scale. It's known that gravity combined with quantum mechanics becomes inconsistent at the Planck scale, if we do it naively. Classical gravity, according to Einstein's theory of general relativity, has to do with space-time. This inconsistency between classical gravity and quantum mechanics means that space-time at very small distances has a different structure than we usually think. It must be not exactly space time-but something fuzzy, something different. It's not clear exactly what it is. That should resolve the inconsistencies.

String theory is a way to resolve most inconsistencies, perhaps not all, but most of them. It's a pretty complex theory that I don't understand because it's not only string theory but also supersymmetry that is used to build a fully consistent theory. But speaking about relation to quantum information, people have realized that gravity has certain general aspects not necessarily related to such elaborate theories like strings. There are some general aspects of the quantum mechanics of space that are applicable not only to space-time, but to other systems as well. At this level, we can ignore or set aside the equations at the micro-scale and think more abstractly. We have a big quantum mechanical system like a black hole, and in this case, gravity kind of counts microscopic degrees of freedom in that system. It doesn't show what those degrees of freedom are. That can be very difficult. But gravity roughly counts them. By looking at the gravitational field, we can know how many degrees of freedom the system has. This aspect of gravity is closely related to quantum information, and people are studying it now.

**ZIERLER:** Finally, last question for today, what does your work at Google tell us more broadly about your research interests and what can and cannot be achieved strictly in an academic setting like Caltech?

**KITAEV:** I view this work as an applied work. They have resources to build a quantum computer, and I'm just trying to participate in that activity. In particular, by exploring new designs of qubits. I don't want to get into details.

**ZIERLER:** I understand, it's secretive and protected. At a general level, though, is this totally different than what Amazon is doing in its partnership with Caltech?

**KITAEV:** It's kind of similar, but it's a competing company.

**ZIERLER:** This has been a terrific conversation. For our next, we'll go all the way back to the beginning and trace your family background in Russia and educational trajectory.

[End of Recording]

**ZIERLER:** This is David Zierler, Director of the Caltech Heritage Project. It's Thursday, January 20, 2022. I'm delighted to be back with Professor Alexei Kitaev. Alexei, it's great to be with you again.

**KITAEV:** Thank you.

**ZIERLER:** In our first conversation, we did a very interesting and wonderful tour of your approach to science and the research. Now, I'd like to go all the way back to the beginning and learn about your family origins in Russia. Let's start with your parents. Tell me a little bit about them.

**KITAEV:** My parents lived in Voronezh. It's the town where I was born. It's actually a pretty big city, now about a million people. My parents were born in different places. My father was born in a little town in Central Russia, and my mother was born in a little town in Ukraine. That was a place where her parents lived for some time. Because her father was in the military service, they traveled a lot.

**ZIERLER:** What were your parents' experiences during World War II?

**KITAEV:** My mother lived in evacuation with her mother in the mother's native city of Orenburg. It's a city east of Moscow. Actually, pretty far east. It was hard for them. It was cold, there was a shortage of food. During that time, my mother's father was fighting in the War. My father's parents were doctors. They worked at field hospitals hospital during the war.

**ZIERLER:** Did your father serve in the military during World War II?

**KITAEV:** Oh, he was too young for that. My grandparents served in the military. My father's parents were doctors, so they served as doctors. My mother's father served as a pilot. Actually, he was a navigator on a military plane.

**ZIERLER:** What were your parents' professions? What did they do for a living?

**KITAEV:** My father has died, unfortunately, but he used to be an applied physicist. He worked at Voronezh University, doing applied physics and engineering, and teaching students.

**ZIERLER:** What kind of physics did he do?

**KITAEV:** He started with radio communications, but in the last 20 years of his life he was doing acousto-optics.

**ZIERLER:** What about your mom? Did she work outside the home?

**KITAEV:** Yeah. She also worked at the university for some time, then she moved to a different place. She was doing an engineering job, but not very high-level. She tried to spend as much time as possible with me and my brother.

**ZIERLER:** Did you see what your father did? Did he take you into work? Did you know what it meant to be a professor of physics when you were a boy?

**KITAEV:** Yeah, it was very interesting. It was a great experience. When I was a little boy, my parents would take me to the university, to my father's lab. I was allowed to play with some equipment. Basically, I was playing with an oscilloscope, looking at the patterns. I could produce Lissajous figures, and that was fun.

**ZIERLER:** Did your father ever talk about politics or the role of the Communist Party in science?

**KITAEV:** Very little. As you can imagine, people's view of politics would be completely different than in the US. You can also imagine that different people had different views. But he didn't talk about politics very much.

**ZIERLER:** You would've been very young, but do you remember the space race between the Americans and the Soviets?

**KITAEV:** Yeah. I guess the time I remember was when the US was launching spaceships to the moon. I remember that there was news that a spaceship orbited the moon and then another spaceship landed with people on the moon. That was when I was a little child.

**ZIERLER:** Were you always interested in math and science?

**KITAEV:** Yeah. Actually, that experience of playing with physics equipment made a great impression on me. I loved science. And I also loved math. At first, I thought I would become a mathematician because I didn't know much of physics, but I was always in the middle. At some point, due to some circumstances, I became a physicist.

**ZIERLER:** Tell me about your elementary school. Was it big or small?

**KITAEV:** It was a big school. It wasn't divided into elementary, middle, and high. It was a ten-grade school. The education was ten grades at the time I was in school.

**ZIERLER:** Looking back, was it a strong curriculum in math and science? You got a good education?

**KITAEV:** Yeah, it was a very reasonable curriculum. In high school, it was specialized in math. I'm not sure if you know this, but in the Soviet Union, there was a system of specialized schools, actually, either in math or foreign languages. In math, it would mean that we would have maybe twice as many math classes as in an ordinary school. Such schools existed in all large cities around Russia. I asked some of my colleagues later, when I was at the Landau Institute, and it turned out they all had gone to such schools. It was a very efficient system to prepare people to work as scientists and engineers.

**ZIERLER:** Did you do very well in school, in math and physics? Were you near the top of your class?

**KITAEV:** Yes, I did.

**ZIERLER:** When you were a teenager, when the Soviets were in Afghanistan, was serving in the military or being drafted something that you needed to deal with?

**KITAEV:** That was a concern. At the time, I was going to the university, and if I failed to be admitted to a university, I would have been drafted. But that didn't happen.

**ZIERLER:** What is the process like of choosing a university in the Soviet Union? What options do you have?

**KITAEV:** We didn't have an application system, where you could apply to multiple universities. You had to choose which one to apply to, go there, submit your documents, and take tests. I decided to go to the Moscow Institute of Physics and Technology, and there were two written tests in physics and math and two oral tests in both subjects. There was also an essay, but that was waived because I received the top grades on the first four tests. Also, I had what's called a gold medal in school, meaning I was excellent in grade school, so they waived the last test.

**ZIERLER:** What year was it when you arrived at the Moscow Institute?

**KITAEV:** It was 1980.

**ZIERLER:** I'm curious, at this point, how much you thought about the Cold War or, becoming a scientist, if you saw yourself as working on behalf of the Soviet Union and what it represented.

**KITAEV:** I'm not sure. I also didn't think much about politics. Of course, I was indoctrinated, like many other people in the Soviet Union. But on the other hand, I just wanted to stay away from politics and do science. I thought that science was useful, of course, and I could do something good. But I was just focusing on the work. [Laugh]

**ZIERLER:** Did you recognize that even just wanting to do science, in the Soviet Union, there were limitations placed on scientists by the government?

**KITAEV:** Actually, science was one of the few things where you could be relatively free from political pressure. I mean natural sciences, like physics, mathematics, chemistry, biology. And that was an advantage of doing science.

**ZIERLER:** Was the Moscow Institute focused exclusively on science and technology, or they offered humanities as well?

**KITAEV:** No, that institute focused exclusively on science. Physics and applied mathematics.

**ZIERLER:** At that point, did you think you would focus on mathematics initially?

**KITAEV:** It's tricky because I first entered the program in applied mathematics, and I was kind of confused because it's called mathematical physics, and I thought I would be doing basically what's called mathematical physics now, meaning doing high-level math and connecting it to physics. But in reality, it turned out to be solving partial differential equations on a computer. They named the program mathematical physics, but in reality, it was focused on solving equations on a computer.

**ZIERLER:** What computers did you have in those early days? What were you working with?

**KITAEV:** Well, I didn't stay at that program very long, but the computer we had access to in the first couple of years at the university was a very old machine. It was controlled by punch cards, so you'd write your code, submit it to someone who'd type the code and produce the punch cards, then you'd run your program. Of course, during the time I studied there, the computers improved a little bit. But those computers were pretty outdated, even for that time.

**ZIERLER:** Had you ever been to Moscow before? What were your impressions when you first arrived?

**KITAEV:** I had been in Moscow, just visiting. Actually, the Institute was outside Moscow in a suburb. I needed to take a train to Moscow. But it was fun to work there and go to places like museums and theaters. It was very interesting.

**ZIERLER:** How big a school is the Institute? Is it like an MIT, a very large program?

**KITAEV:** I think it's smaller than MIT but bigger than Caltech. While I was there, one class had 100-plus people, and there were several departments. About 700 students times six because there were six years of study. Between 4,000 and 5,000.

**ZIERLER:** After your initial foray into mathematical physics, what program did you settle on?

**KITAEV:** I switched to theoretical physics in the third year. And I needed to take an exam to enter the group. I started doing theoretical physics from year four.

**ZIERLER:** This was more comfortable to you? You felt more at home in theoretical physics?

**KITAEV:** Actually, I failed the exam the first time. But eventually, I succeeded. [Laugh]

**ZIERLER:** What was most interesting to you when you first got involved in theoretical physics? What were you working on?

**KITAEV:** I was in Lev Gor'kov's group specializing in condensed matter. At the Landau Institute, there were faculty doing all kinds of theoretical physics. There were some students doing different things, but most of the students specialized in condensed-matter physics, and I also worked in that field. When I decided to go there and take the test, I read something about condensed-matter physics, and I was excited. In particular, I read Anderson's paper explaining the results for which he got the Nobel Prize. It was about spin impurities in metals. It was fascinating because he proposed a simple model, which explained some experimental facts. This way of thinking, just inventing models that don't have to be very complicated but must capture essential things about the physical system, fascinated me. Throughout my career, I've focused on inventing simple models.

**ZIERLER:** I'm not sure how the names translate in Russian, but around this time, there was the switch in the United States from the term solid-state physics to condensed-matter physics. Was there a similar transition in the use of terms in Russia?

**KITAEV:** It was called solid-state physics.

**ZIERLER:** Did you stay at the Institute for graduate school? How did that work?

**KITAEV:** I stayed at the Landau Institute, but let me explain some details, which might sound unusual. It was a unique system. The Moscow Institute of Physics and Technology provided general education, mainly in the first three years. Then, the curriculum became more specialized in various fields of physics and applied mathematics. Those courses were offered at research institutes. People working at research institutes in Moscow in the suburbs had a dual affiliation in their lab at the Moscow Institute of Physics and Technology. They were teaching students, either giving courses or doing lab work at those research institutes. I was taking advanced classes at the Landau Institute of Theoretical Physics for the last three years of my study. Then, I switched to the graduate school at the Landau Institute.

### Quasicrystals at the Landau Institute

**ZIERLER:** Tell me about the Landau Institute. What was it known for?

**KITAEV:** It's an institute of theoretical physics. It was named after the famous physicist Lev Landau, but it was established by Landau's students, including Khalatnikov, the first and longtime director, Gor'kov, Abrikosov, Dzyaloshinskii, and others. It was a great place for theoretical physics. At the time I came, there were about 80 physicists doing completely different things. If you had a question in theoretical physics, practically in any area therein, you could find an expert to talk with.

**ZIERLER:** How international was the Landau Institute? Were the professors traveling abroad? Would there be visitors from other countries who came to the Landau Institute?

**KITAEV:** Yes, they traveled abroad. In the Soviet Union, traveling was rather difficult, but they still traveled and had visitors. The first time I came there was in 1984 as a student. During the initial years, when it was still the Soviet Union, traveling was limited, but we still had some foreign visitors who would give talks. That was great. Later on, many people went abroad and just stayed there. After that, the Landau Institute was not so great anymore. That happened in the early 90s.

**ZIERLER:** Who was your advisor at the Landau Institute?

**KITAEV:** Valery Pokrovsky.

**ZIERLER:** What did he work on?

**KITAEV:** At the time, he worked on the fractional quantum Hall effect. At least, that was the topic I was trying to work with him together on. But somehow, it didn't happen, and I did something completely different. He's known for his work on phase transitions and was doing some related work even at that time.

**ZIERLER:** What about for you? What did you want to focus on, what was interesting for your thesis research?

**KITAEV:** I found a topic on which I worked with two of my fellow students, my friends, Leonya Levitov and Pasha Kalugin. [The full names are Leonid and Pavel.] The topic was quasicrystals. We were lucky that as students, we were learning incommensurate systems. Let me explain what quasicrystals are, how we learned about them, how they're related to incommensurate systems. Crystals have periodic arrangement of atoms. A periodic array of atoms can have certain symmetries, like cubic symmetry, but they cannot have, for example, icosahedral symmetry. This is easier to understand in two dimensions. You cannot have a periodic lattice with a symmetry axis of order five. For example, you cannot pack pentagons in a regular periodic pattern, but that is a special case. It's a mathematical fact that there's no lattice with pentagonal symmetry. Of course, that means that there are no crystals with pentagonal or icosahedral symmetry. In 1984, there was a discovery. Experimentalists saw something that looked a crystal with an icosahedral symmetry. The electron diffraction pattern showed Bragg peaks like a crystal. But their arrangement had clear pentagonal symmetry. That was done by experimentalists. We saw that paper, and we came up with an explanation that it's sort of an incommensurate structure. The problem with icosahedral symmetry is that it's not possible to have only three periods in the X, Y, and Z axis such that their integer combinations would have an icosahedral symmetry. But it is possible to have six independent periods. That means that the structure can be pictured as a periodic lattice in six dimensions and then restricted to a three-dimensional subspace. Secretly, it's six-dimensional. That was our first paper on quasicrystals. We studied quasicrystals for several years. I did my PhD on that subject.

**ZIERLER:** Were you aware of Paul Steinhardt's research? Was he working on quasicrystals at this point?

**KITAEV:** Yes, he was. We read his paper. First, we came out with our explanation, then we saw his paper with Levine. His explanation was better in the sense that it was geometric. Basically, it was based on Penrose tiling. In Penrose tiling, you can force this quasiperiodic order using local constraints. Our explanation used an effective free energy. For metals, actually, our explanation works well because interactions in metals are not short-range. I still don't know which explanation is better.

**ZIERLER:** As a graduate student, were you allowed to travel abroad? Could you go to international conferences?

**KITAEV:** As a graduate student, I could actually travel. That was a change in the last few years of the Soviet Union. It was nice. I went to Trieste to attend a conference on quasicrystals. I also went to India shortly after my PhD.

**ZIERLER:** Was your thesis research on quasicrystals connected to condensed-matter experimental physics or observation in any way?

**KITAEV:** Of course, it's condensed matter. Our work was related to the experiment by Shechtman, Blech, Gratias, and Cahn. We explained the structure of a quasicrystal. It wasn't very closely connected to experiment because it's kind of a general explanation.

**ZIERLER:** What does the term quasi mean? Why a quasicrystal?

**KITAEV:** Because it has some periodicity, but it's not exactly periodic. There is a definition of a crystal that says it's a pattern that has translational symmetry. In three dimensions, it's three independent translations. A quasicrystal doesn't have a translational symmetry, but it still has some periodic motives, which are seen in diffraction patterns.

**ZIERLER:** To foreshadow a little bit, in graduate school, do you see intellectual origins that would lead to your interests in quantum information? Is there a natural intellectual connection that got you from condensed-matter theory to what would come to be known as quantum information?

**KITAEV:** There is no intellectual connection with condensed matter, but there was something that sparked my interest in quantum computation, and it happened very early while I was still an undergraduate. I read a book by Yuri Manin, *Computable and Uncomputable*. The book is on mathematical logic, but it had an introduction where Manin tried to come up with some examples where logic could be useful. It's not trivial to come up with such examples because logic is a very abstract subject. One idea was that simulating quantum mechanics is computationally difficult. Manin thought that might be an indication that quantum mechanics itself can work as a powerful computing assistant. Two paragraphs in the introduction caught my attention. It was an expression of this idea.

**ZIERLER:** Why did that catch your attention? What was so interesting to you about this?

**KITAEV:** I realized it was a great idea. I didn't know whether it was novel enough, but it was really great because quantum mechanics was mysterious to me. I didn't know enough quantum mechanics. It was during my second year of college when I started reading Feynman's lectures on quantum mechanics. Feynman has some elementary topics in quantum mechanics as part of his famous lectures. I hadn't studied quantum mechanics seriously at that time, so it was a kind of mysterious and fascinating subject. Then, I learned that quantum mechanics could be used to do computation in principle. But I couldn't work on that idea right after reading Manin's book. I started working after I had read about subsequent developments. First, I read Feynman's paper in 1985.

**ZIERLER:** Which one is that?

**KITAEV:** He wrote two papers on the connection between quantum mechanics and computation in the early 80s, and the second was in 1985. I don't remember the titles of those papers, but in that second paper, he talked a bit about quantum circuits implicitly. He didn't have a definition of quantum circuits.

**ZIERLER:** He imagined what quantum circuits would be.

**KITAEV:** Right. he had some ideas how a quantum mechanical system could be used to compute. Later, I learned some developments in quantum algorithms. Then, the whole program made sense to me. I knew how to make progress. Another thing that was helpful to make progress is, I studied some computational complexity theory. Because quantum computation is on the boundary between physics and computer science, you also need to know the other side. You need to know about equivalencies between computational models because one could come up with various ways of using quantum mechanics. For example, one could try to create a quantum mechanical system and measure its energy levels with high precision. That would be also something computational. It's important to realize that you need to come up with just one way of doing computation, and other ways can be reduced to it. I got this idea in concrete examples from studying computational complexity. By the time I studied the subject, there was a circuit model of quantum computation, and it was clear how to simulate quantum systems using that model. Of course, at that time, formal results had not appeared yet about such simulations. Much later, John Preskill described how to simulate field theory. But conceptually, the circuit model clearly allowed the simulation of some other things, and it felt like it was the right model.

**ZIERLER:** To be clear, all of the things that you're talking about, you started thinking about after your PhD, or even during graduate school?

**KITAEV:** It was during my graduate school. I was just occasionally learning some new things. But I actually started working on this subject a few years after my PhD, around 1995. Just as I started, I learned about Shor's algorithm. Well, first I heard that Peter Shor found a solution to the discrete logarithm problem using a quantum computer. I thought for some time and also found a solution. That led to my paper about eigenvalue measurement.

**ZIERLER:** By this time, in graduate school, how was your English? Are you reading papers in English? Were they translated in Russian? How did that work?

**KITAEV:** I read papers in English. My English was not very good. I was trying to improve it. Actually, I was studying some English in grad school, and that helped. But of course, the education was such that we were supposed to know English and read papers as undergraduates. I could read papers. I couldn't speak English very well as an undergrad. During grad school, I took additional classes to improve my English.

**ZIERLER:** Of course, this is before the internet. How long did you have to wait until papers or journals made their way to the Soviet Union?

**KITAEV:** Some journals appeared in some libraries pretty quickly, but it still would take a month or so. The Institute's library was subscribed to *Physical Review Letters* and *Physical Review B*, which were delivered promptly, but I remember checking for other journals in other libraries and waiting for them to arrive.

**ZIERLER:** Is the process for defending a dissertation in the Soviet Union more or less similar to the United States, where there's a committee and you defend?

**KITAEV:** Yes, you defend a dissertation. There is also a committee. At the Landau Institute, you needed to present your work at a seminar before the dissertation. Then, the dissertation is a short thing. It's kind of formal because you give a very short presentation of several results. There isn't actually time for people to follow and have a serious discussion during the dissertation. But the candidate is supposed to present before that at the regular seminar.

**ZIERLER:** The year you defended was 1989?

**KITAEV:** Yes.

**ZIERLER:** The big political changes that were happening at that point, the fall of the Berlin Wall, were you thinking about that and what it might mean for your career and opportunities?

**KITAEV:** Of course, I was thinking about that, but I didn't know how it was going to end up. It was exciting that people got more freedom. Of course, the Berlin Wall is the Western perspective. We didn't think about the Berlin Wall in the Soviet Union. What it meant was that people had more freedom, but early on, it was becoming clear that some bad changes were going to happen. I didn't realize they would be so bad. Inflation started, and it was difficult to save any money.

**ZIERLER:** When you're talking about freedom earlier, are you referring to *Glasnost* and *Perestroika*?

**KITAEV:** Yeah. I think around 1987, Perestroika started. People received some political freedom. It became easier to travel. And actually, I took advantage of that and visited some countries. As a graduate student, I made only one trip to Trieste in Italy. The other trip happened after I graduated and got my PhD. But still, it was a time when traveling became easier than before.

**ZIERLER:** I mentioned the Berlin Wall because, if I understand correctly, ironically, Mikhail Gorbachev instituted *Glasnost* and *Perestroika* in order to sustain the Soviet Union, to keep it together. Of course, that was the process that helped break it apart. After the fall of the Berlin Wall, did you feel like the Soviet Union was going to collapse?

**KITAEV:** No, I didn't have this sense. I didn't think about foreign politics very much. Now, people recall what happened, that Gorbachev gave up something to the West. The influence of the Soviet Union over Eastern Europe. I didn't think about that. I was thinking about life in the Soviet Union. The sense was that people could do more. But the feeling of something bad looming started with inflation. Eventually, the situation became quite bad economically. You may know that there was a big crisis starting in 1990 and continuing for a decade at least.

**ZIERLER:** At least intellectually, after you defended the dissertation, were you thinking more about computation and quantum mechanics, that perhaps being the trajectory of your career, or were you still focused on condensed-matter theory as your academic specialty?

**KITAEV:** I was just trying to do something that interested me. I still do the same now. I switched from subject to subject occasionally. I cannot switch very far, like to biology, but within physics. [Laugh]

**ZIERLER:** What professional opportunities did you have after you defended, particularly because this was right when the economic crisis started to happen?

**KITAEV:** I was lucky that the director of the Landau Institute, Isaak Markovich Khalatnikov, arranged a program together with the Weizmann Institute, where scientists from the Landau Institute could visit. I visited twice, in 1991 and 1993, working for half a year each time at the Weizmann Institute, and that helped me financially. It was a very difficult time in Russia. I continued working in this mode, having some long-term visits to foreign countries. The other country was France, which I also visited twice for a long time, working at IHES. This way, I could earn enough money for a living in Russia because just staying at the Landau Institute and receiving the salary was not enough. There was one time, and I should acknowledge that I'm grateful for this to the Landau Institute administration, that they gave me a prize and raised my salary for a year substantially because of my work in quantum computation.

**ZIERLER:** What was your position at this point? Was it a post-doc, essentially?

**KITAEV:** The system is different. There's no official post-doc position. Maybe now, there is, I'm not sure. After my PhD, I was hired as a junior member of the Landau Institute. That was a permanent job in the sense that there wasn't a fixed term. But there wasn't a legal protection either, like tenure in the US. It's just a job that is, by default, permanent. People often stayed for many years and got raises in salary. There's no sharp distinction between junior members and more senior positions.

**ZIERLER:** Tell me more about these international visits you were able to go on. For example, in France. What were you working on at that point?

**KITAEV:** In France, I was working on quantum chaos. I started doing it with Uzy Smilansky when visiting the Weizmann Institute. Quantum chaos could mean different things. At that time, the popular subject was energy level statistics and how it's related to properties of classical trajectories. At this time, I would say quantum chaos is about out-of-time-order correlators. This way, one can close a parallel with classical chaos. But I started working on quantum chaos in 1993, I think, together with Uzy Smilansky. Then I realized that I wasn't making much progress on quantum chaos per se, but I could make some progress understanding some related mathematical problems, so-called dynamical zeta function, which was invented by Ruelle and studied by other people. There are several variants of the zeta function. I showed that a particular variant is an analytic function.

**ZIERLER:** What does that mean?

**KITAEV:** It's a function of a complex variable. It's holomorphic in a certain disc that depends on the smoothness of the dynamical system. If the dynamical system is defined by smooth equations, then the disc extends to the whole plane. I mean, the function is holomorphic in the complex plane. It's a technical thing. I was working on this type of problem when visiting France the first time. By the second time, I think, I had switched to quantum computation.

**ZIERLER:** Do you have a clear sense of when the term quantum computation or quantum computer started to be used in the sense that we're doing science that might eventually lead to a quantum computer, not just a quantum computer as a theoretical concept?

**KITAEV:** It was assumed that once we developed the theoretical concepts, they would be implemented. Maybe not immediately. I don't remember.

**ZIERLER:** Did the experience of traveling abroad plant a seed in you to think that you might want to leave Russia someday and emigrate?

**KITAEV:** I didn't want to emigrate until I realized that working at the Landau Institute was not great anymore, and I didn't have enough opportunity to discuss my work with other people at the Landau Institute. I was the only person working on quantum computation. Also, the regular salary at the Landau Institute was not enough for a normal life. Not enough to support a family. I would have to earn money somewhere else anyway. These days, people have grants, Russian or sometimes European grants. I'm not sure how European grants work right now, maybe due to the political situation, they don't work anymore. But at least this was the way for people to get sufficient money, and it's a lot of hassle, applying for those grants. Maybe at the time I left, it wasn't even an option. I left in 1998. Russia was very poor at that time. I thought that wasn't even an option, that people were just poor in Russia, and that's it. The only way to make any money was to go abroad. Of course, there were some professions in which you could earn money in Russia, but they were not related to science. I didn't want to immigrate until I realized that my prospects staying in Russia were not very good.

**ZIERLER:** What context did you have? What availability did you have to establish connections to see if you could have an opportunity beyond Russia?

**KITAEV:** I wasn't actively seeking an opportunity to immigrate. In hindsight, it was smart because when I made this decision, I was already known as a physicist. I visited Caltech and IBM in 1997, then I made a long-term visit to Caltech in 1998. During that visit, I decided to stay in the United States.

**ZIERLER:** What was that initial point of connection with John Preskill? Did he write to you? Did you get in touch with him?

**KITAEV:** He wrote to me. We also met at a conference in Turin on quantum computation. He first invited me, and I visited in 1997, then came the conference, where he made an offer for me to visit for a longer time.

**ZIERLER:** What were you working on specifically that caught John Preskill's attention at that point?

**KITAEV:** I was studying anyons and doing quantum computation by anyons.

**ZIERLER:** John Preskill told me in wonderful detail how important this was at the time. Looking back, how did that contribute to the questions John was considering at this point?

**KITAEV:** John studied anyons as a field theorist. He had some papers about anyons and related objects in higher dimensions. I remember one paper was called *Alice Strings and Cheshire Charge*. Those papers studied possible anyons in high-energy physics. I think I was the first to realize that anyons are connected to quantum computation.

**ZIERLER:** What was that insight? How did you make that connection?

**KITAEV:** I knew something about anyons in fractional quantum Hall systems. I remember that studying the fractional quantum Hall effect helped a lot. I came up with the idea of the Toric code using fractional quantum Hall as an analogy. My first thinking was to put the fractional quantum Hall system on a torus. Somehow, I realized that the ground state would be degenerated and protected from errors if one puts the fractional quantum Hall system on a torus. I didn't know at the time, but the degeneracy actually had already been established by Einarsson in this geometry. But the idea that it would be protected from errors was original, I think. I believe that people had not discussed that before.

**ZIERLER:** Did you and John see this as complementary to what John was working on? Did it expand on his ideas? Did it move the field forward in a way that he was not doing? How did all of these things come together as an intellectual collaboration?

**KITAEV:** I don't remember exactly. We had some discussions. I can't speak for John, but from my perspective, it was interesting that this concept that was born in high-energy and condensed-matter physics found an application, and it seemed very natural to me that anyons and topological states are useful for quantum computation because they offer protection from errors. Discussing with John, we talked about whether nature works in the same way. I recall that we had such a discussion. Whether nature is fault-tolerant.

**ZIERLER:** When you first came to Caltech, was that as a visitor? Did John bring you out here to give a talk?

**KITAEV:** The first time, not exactly to give a talk but to visit for a month and have some discussions.

**ZIERLER:** So a faculty appointment was premature at this point.

**KITAEV:** Right. The first visit was for a month. The second visit was for nine months. After that, I went to Microsoft Research in Redmond and spent two years there. Then, John was able to offer me a research position at Caltech funded by his grant and IQI. It was still called IQI at that time. A year later, I got a faculty position at Caltech.

**ZIERLER:** What were your impressions of Caltech during that initial first month? What did you think of it?

**KITAEV:** It was very nice. It was nice discussing things with people because at the Landau Institute, I didn't have this opportunity. As I said, I was the only person working on quantum computing at the Landau Institute. At Caltech, I could discuss with John Preskill and his students. Daniel Gottesman was there, and I was discussing quantum computation with him. But I don't remember whether it happened during my first one-month visit or later, but that was fun, too.

**ZIERLER:** You were there for one month, then you went back to Russia, then you returned for nine months?

**KITAEV:** Yeah.

**ZIERLER:** Did John share with you his vision at that time of creating what would become the IQI, that he wanted to formalize this into a proper institute?

**KITAEV:** I don't remember.

**ZIERLER:** What was your subsequent appointment for nine months? A visiting professor essentially?

**KITAEV:** Yes. It was a research position, and I also gave some lectures on quantum computation.

**ZIERLER:** In what ways was being in that environment not like at the Landau Institute where you were all by yourself? How did talking with John, the graduate students, the post-docs influence your thinking about quantum information at that point? What were some new ideas you were getting exposed to?

**KITAEV:** I think it was on the technical level. I learned about other ways to do error correction. I had a scheme for doing error correction, and people here, in particular, Daniel Gottesman, came up with other ways of doing error correction.

### From Station Q to Caltech

**ZIERLER:** Did the opportunity at Microsoft come together while you were a visitor at Caltech? Or did you go back to Russia and later secure this position?

**KITAEV:** I went back to Russia, but I got this position while working at Caltech. In addition to this nine-month offer from Caltech, I also got an invitation from Michael Freedman to visit Station Q. I talked with Michael, and that was fascinating in its own way because I learned some mathematics. I like mathematics.

**ZIERLER:** What was the connection between you and Michael Freedman? Who introduced you?

**KITAEV:** Michael emailed me. The reason he did was because he had the idea of using topology in quantum computation. Then, he learned about my work on anyons. By the time I got the invitation, I think the paper had already appeared on the arXiv. That's why Michael got interested in my work, because it connected to his idea. I had a more concrete realization of how to compute using topology, but the general idea was similar.

**ZIERLER:** Tell me about Station Q. How well-developed was it at that point?

**KITAEV:** Station Q had not started yet. It was Microsoft Research at Redmond. It was run by Jennifer Chayes and Christian Borgs. It was basically a group of mathematical physicists, supported by Microsoft, who were doing the usual stuff. Michael accepted this position coming from UC San Diego. But it was his dream to make something focused on quantum computation. He founded Station Q a few years later.

**ZIERLER:** What was the excitement at this point at a place like Microsoft? Was the vision that Microsoft really was going to create a quantum computer and that it would be useful for what Microsoft wanted to do in the future?

**KITAEV:** No, I think at that time, certainly before Station Q, it was just some abstract subject. It wasn't the idea that we were doing something practical. Station Q was founded with the idea that a quantum computer would be built eventually. But at that time, in 1998, I think it was just a topic in mathematical physics.

**ZIERLER:** Who else besides you and Michael were at Microsoft and thinking about these things?

**KITAEV:** I mentioned Jennifer Chayes and Christian Borgs. László Lovász was also working at Station Q at that time.

**ZIERLER:** But it was a real research group, there were several people there.

**KITAEV:** Several people. It was a small group.

**ZIERLER:** Did you stay connected to Caltech? Were you following developments with the IQI?

**KITAEV:** I stayed somewhat connected. I had some communication with John. We wrote a paper with him and Daniel Gottesman about encoding a qubit into an oscillator.

**ZIERLER:** At this point, what is the work situation? Are you on a visa? Do you see this as a temporary opportunity, then you'll go back to Russia? Or did you see this as the start of making a new life for yourself in the United States?

**KITAEV:** I decided to stay in the United States during my first year at Caltech. I wanted to stay in the United States because it was clearly a better environment than the Landau Institute at that time, at least for me.

**ZIERLER:** When you say the Landau Institute, I wonder if you can extrapolate to all of Russia. Were there any programs in Russia focused on quantum information at this point?

**KITAEV:** I don't think so. But I should say that at the time I came to the Landau Institute in the 1980s, it was one of the best places in the world for theoretical physics.

**ZIERLER:** Right. Just not quantum information.

**KITAEV:** Well, quantum information didn't exist at the time. But there was a pretty sharp and unfortunate transition in the early 1990s that led to this decline at the Landau Institute.

**ZIERLER:** What were the topics or big ideas you were focused on during your time at Microsoft?

**KITAEV:** At Microsoft, I continued to think about anyons and topological quantum computation. We did some work with Michael Freedman, Zhenghan Wang and Michael Larsen on topological quantum computation.

**ZIERLER:** Was this a temporary position, a staff position? Could you have stayed longer at Microsoft if you'd wanted to?

**KITAEV:** No, it was a temporary position for two years. They didn't offer me an extension. Later, I was offered a position at Station Q. But I decided to return to Caltech. I spent a year at Station Q.

**ZIERLER:** What year was that?

**KITAEV:** Around 2005.

**ZIERLER:** Tell me about the process that got you back to Caltech.

**KITAEV:** First, I returned to Caltech in 2001 as a research scientist. I'm grateful to John for offering me this position. It was funded by IQI, and then Caltech considered me as a faculty candidate and offered me a permanent position.

**ZIERLER:** Were you collaborating with John? For example, the paper *Topological Quantum Memory* came out in 2001. Would this have been a collaboration with John and others while you were at Microsoft, or this was work when you were a research fellow at Caltech?

**KITAEV:** The paper was written when I had just returned to Caltech. We may have started the work while I was still at Station Q, I don't remember.

**ZIERLER:** Just a question on the nomenclature, that's a new term that we haven't discussed. What does topological quantum memory mean?

**KITAEV:** Quantum memory is a way to store quantum information, to store qubits. To store them reliably, one needs some error correction. Topological quantum memory is a way to protect those qubits from errors using topological properties of physical systems. In that particular paper, we discussed a surface code, which is not exactly a physical system. It can be realized at the logical level using more conventional physical elements. It's an error-correcting code, but we discussed error correction from the perspective of statistical mechanics. We showed that the probability of an uncorrectable error can be expressed as a statistical mechanics problem. We also considered some algorithms for correcting the errors.

**ZIERLER:** Was this related at all to a similar project you were working on at the time, Wilson loop operators?

**KITAEV:** That was a different project with John Preskill.

**ZIERLER:** What was different about it?

**KITAEV:** Loop operators was John's idea. I made some contribution to that work, but mainly, it comes from John's original work on field theory, the question whether a Wilson loop operator can be measured quickly. In that paper, we explored causality constraints on what can be measured quickly because one cannot communicate faster than light, and one result was that a non-abelian Wilson loop cannot be measured quickly. One actually needs to go around this loop and collect information. That was John's idea entirely. I came up with a different thing about abelian Wilson loops, which can be measured fast.

**ZIERLER:** Fast relative to what?

**KITAEV:** You don't have to actually go around, and if you calculate the time the light needs to travel around this loop, it's not the measurement time. The loop operator, or the magnetic flux through the loop, can be measured much faster. But some preparation is needed. One needs to prepare for the measurement in advance, and then you can do it quickly.

**ZIERLER:** One last question for today. When you returned to Caltech, this was during the transition period. You were there as a researcher as John Preskill was starting to think about the IQI. When you returned, the IQI was just getting started, already in existence. In what ways had quantum information grown institutionally at Caltech? In what ways were there more graduate students, more post-docs, a greater collaboration in working on these ideas?

**KITAEV:** Quantitatively, certainly, there were more people supported by IQI over time. It was a bigger group. John ran group meetings where the students would speak. There was also a regular seminar on quantum computation. It was just a sufficiently big group of people to have a seminar. It was very good. But I cannot tell much because I kind of worked alone or collaborated with certain people, mostly with John. Somehow, I don't have much interaction with people in general. Only when we work on a particular problem.

**ZIERLER:** Well, we'll pick up next time when you join the faculty in 2002.

[End of Recording]

**ZIERLER:** This is David Zierler, Director of the Caltech Heritage Project. It's Thursday, February 3, 2022. I am delighted to be back with Professor Alexei Kitaev. Alexei, it's good to be with you again.

**KITAEV:** Thank you.

**ZIERLER:** Today, I'd like to start with some of the research you were doing in the early 2000s. We talked in our first discussion about Majorana fermions. You wrote a paper on unpaired Majorana fermions in quantum wires. What are quantum wires?

**KITAEV:** A quantum wire is essentially a one-dimensional system. It can be made out of a semiconductor such that the transversal motion is quantized, and effectively, electrons only move in one direction. But at the time, I didn't know how to make such a wire. It was an abstract model, a Hamiltonian that has certain interesting features.

**ZIERLER:** You say, "At the time," what progress has been made in the field since 2000?

**KITAEV:** Since 2000, other people, including Gil Refael at Caltech, figured how to actually make such wires with existing materials. One needs a combination of three properties, which I knew at the time, but I didn't know about concrete examples. It should be a system with a strong spin-orbit interaction. There must be a superconductor; if it's not the wire itself, it should be in proximity to that wire. Third, there must be a magnetic field, something that breaks the time-reversal symmetry. It is usually broken by a magnetic field.

**ZIERLER:** That same year, along with Daniel Gottesman and John Preskill, you wrote the paper *Encoding a Qubit in an Oscillator*. In what way did that paper reflect some of the new thinking about error-correcting codes?

**KITAEV:** It's a different type of an error-correcting code. The usual error-correcting codes are based on qubits. One qubit is encoded into many. One can encode a qubit into a quantum many-body system, also, but this code is an encoding in a pretty simple quantum mechanical system, a harmonic oscillator. It's not a qubit, it has multiple states. But yeah, it's a new type of code. It has been implemented recently by Devoret's group at Yale.

**ZIERLER:** Which gets us where? What's the progress there?

**KITAEV:** They figured out how to do the stabilizing measurements on the quantum state. The quantum state is realized using an electromagnetic resonator. The measurements are done by interaction of this resonator with a superconducting qubit based on Josephson junctions. They can store a qubit for some time. It's pretty short, only slightly better than without encoding. One can hope that things will improve.

**ZIERLER:** Tell me about some of your collaboration with Michael Freedman at Microsoft. What were you working on with him in the early 2000s?

**KITAEV:** We were working on topological computation. Michael Freedman had a similar idea to mine about the same time as I wrote my paper, in 1997, about quantum computation with anyons. We were working on various kinds of anyonic computation, and in particular, we were interested in simulating the general quantum computer using other topological systems. Well, anyons are particles that exist in a fixed geometry. Say we create a two-dimensional system, it lives on a plane, and we can excite the system. The excitations are anyons, and we can drag them, move them around, and do a computation. But that requires pretty sophisticated anyons. Another way of doing computation, if we can change the geometry or topology of the system itself, if we can, say, cut holes, connect different holes with bridges, and things like that. We were studying both settings, and we were proving results like equivalence between the standard model of quantum computation and the topological model.

**ZIERLER:** Tell me about your motivations in writing, along with Alexander Shen and Mikhail Vyalyi, the textbook *Classical and Quantum Computation*. Why did the field need an introductory book that looked at both classical and quantum computers at this time, 2002?

**KITAEV:** First, at that time, there weren't many books on quantum computation, so a quantum computation book was certainly needed. Actually, the Russian version of this book appeared earlier, in 1999. The original version was based on lectures I gave at the Independent Moscow University. It was a lecture course, then it became a book in Russian. The English version included more material.

**ZIERLER:** Where did you know Alexander and Mikhail from?

**KITAEV:** Mikhail Vyalyi is my friend, he's still in Moscow. I also knew Alexander Shen, a computer scientist who now works in France. At the time, we were all in Moscow, and I attended a complexity theory seminar on a regular basis, and we usually met there and talked. It was a community I belonged to.

**ZIERLER:** And why did the book include information on classical computing? Why was it important to survey classical computing and compare that with quantum computing? What was the value in that?

**KITAEV:** It's still my conviction today that classical computation is a prerequisite for quantum computation. To understand what quantum computation is, you need to know quantum mechanics, but you also need some concepts from complexity theory. It's important to understand that there are different models of computation, and all these models are equivalent in the sense that all computers can compute exactly the same things. There are some differences in the complexity of computation, one model may be more efficient than another. All natural classical models are equivalent up to polynomial factors, and more often, even up to logarithmic factors. When we add quantum elements to it, one can, in principle, achieve an exponential speedup. It's important to have a general theory that includes models and transformations between models. That's the purpose of the classical part of the book.

**ZIERLER:** We talked about quantum protocols previously. Your work with Preskill and Dominic Mayers, superselection rules and quantum protocols. What are superselection rules?

**KITAEV:** Superselection rules are a concept related to anyons or other particles. The name comes from a previous concept called selection rules. Selection rules in quantum mechanics are rules saying that certain transitions are allowed by symmetry, and other transitions are not allowed. They are important in atomic physics, for example. When an atom emits a photon, it cannot go from any level to any other level. There are some constraints, and these are called selection rules. Now, superselection rules are, in a way, similar, but deal with processes between particles, what can happen when particles collide and produce other particles. These constraints are often related to gauge symmetry, which is different from global symmetries that are relevant to atoms. But one can also say that the superselection rules are actually more fundamental than gauge theory. Gauge theory is just a way to represent these rules. In a given system, excitations, which we view as particles, may have some constraints, like the electric charge is conserved. In anyonic systems, we can see that, for example, a Z2 charge can be conserved. We talked about surface codes that are built of qubits interacting with each other in two dimensions. This is a very artificial anyonic system. But such systems could be built from other elements. It's not known how to do this well, but in principle, they could be built from electron spins.

These are systems where there's no preexisting selection rule of conservation law. One can do anything with spins, all symmetries can be broken. Yet, when the spins interact, they form an entangled state, and this is the ground state, or people call it "vacuum" because it's analogous to the vacuum as understood in high-energy physics. It's not just an empty space, it's actually a more complex thing. This entangled state of spins can be in the ground state or in an excited state. Excitations are quasi-particles, and in this particular system, those quasi-particles are conserved modulo 2, so that you cannot produce a single quasi-particle. When you do anything with this system locally, not destroying the whole thing but just changing it locally, you can produce two particles, but not one.

**ZIERLER:** In what way did this research suggest that there was interesting work happening in quantum cryptography at this point?

**KITAEV:** Actually, I don't remember this work so well. Somehow, I'm not so interested about cryptography. I don't remember.

**ZIERLER:** Do you have a sense of when the National Security Agency started to get concerned about quantum computers and internet security?

**KITAEV:** Well, they may be concerned that another country like China will make a quantum computer. That could be a concern. But quantum computers will not be available to many users for a long time. I don't know if and when the NSA started to be concerned. They should be concerned now because when quantum computers are made, it will take time to change the codes, and this has to be done everywhere, including consumer systems. It's a long process. Some codes already exist that we don't know how to break on a quantum computer. One needs to find efficient variants of those codes, write software, test it, deploy it, so it's a long process. I suppose they are concerned, but I don't know.

**ZIERLER:** Your work with Sergey Bravyi in 2005 on universal quantum computation–what does that mean? What is universal quantum computation?

**KITAEV:** Universal means you can compute anything. There are some restricted models of computation, where you can do some operations, but you cannot really do much. These models are still interesting to study some properties of quantum systems. They're interesting mathematically. But eventually, we want universal computation such that it can compute anything.

**ZIERLER:** What does that mean, compute anything?

**KITAEV:** You can compute any function. You have some inputs and outputs, and for example, you can add two numbers, you can multiply numbers, you can split an integer into prime factors. Any of these algorithmic tasks are doable. In a classical computer, they're all doable if you have a certain set of boolean gates. For example, if you have the AND, OR, and NOT gates, you can do anything. If you only have the AND and OR gates, but no NOT gate, then there are some restrictions on what you can compute.

**ZIERLER:** That same year, your work on universal manifold pairings and positivity, another question on universality. What does that mean, universal manifold pairings?

**KITAEV:** Pairing between manifolds means that we glue two manifolds with boundary along that boundary and construct a closed manifold. We can glue two semi-spheres to make a sphere. There are invariants of three-dimensional manifolds where a 3-manifold with boundary defines a quantum state associated with that boundary. When we glue two manifolds, we construct the inner product between the corresponding states. This pairing between Hilbert spaces depends on a topological theory. There is a way to associate Hilbert spaces with two-dimensional manifolds and vectors with three-dimensional manifolds whose boundary is the two-dimensional manifold. But such a theory is not unique. There are many ways to do so. Each such theory defines a new pairing, a new inner product. What we studied in that paper is the general inner product between manifolds themselves. That was the idea of universality. My contribution to this work was very minor. It was Michael Freedman's project.

**ZIERLER:** We've talked a lot about anyons. A sole-author paper in 2006, *Anyons: An Exactly Solved Model and Beyond*. What is an exactly solved model?

**KITAEV:** When we consider a classical or quantum model, it's defined by a Hamiltonian, and we may or may not be able to write the solution to this model, how the system evolves and what the energy states are. We may or may not be able to do that in a closed form as an analytical expression. And there are somewhat technical definitions of exact solvability. This is an example from classical mechanics. One can solve the two-body problem with Newton's law of attraction. The earth moves around the sun along an elliptical orbit. Everything can be written in a closed form. With three bodies, there's no such simple solution. In quantum many-body physics, we can consider models with spins or electrons, and only a small fraction of models are exactly solvable. In that paper, I described a new model, actually a pretty interesting one because it's a spin model, it's not based on symmetries. There are some symmetries, but the model features anyons that are not related to those symmetries. Depending on parameters, those anyons can be abelian or non-abelian. There is a bunch of interesting properties. I want to say that right now, people are studying some possible experimental realizations of this model. The experimental picture is not very clear because one cannot make the exact same interaction in practice, but one can come close. There are some candidate materials that seem to be described by the same Hamiltonian and might also harbor anyons, but that has not been confirmed.

**ZIERLER:** On the question of abelian and non-abelian, that same year, you worked on non-abelian statistics in the fractional quantum Hall state. When did you become involved in this research?

**KITAEV:** Actually, non-abelian statistics in the fractional quantum Hall state was studied a long time ago by Moore and Read. They were the first to discover non-abelian anyons in the quantum Hall setting. That was sometime in the early 90s. I don't remember my work on this subject. [Laugh]

### Preskill and Topological Entanglement Theory

**ZIERLER:** What was some of the work you were doing with Preskill on topological entanglement entropy at this point?

**KITAEV:** If there is a system with anyons, and we cut a disc out of that system, one can study the entanglement between that disc and the spins around it. The system is made out of spins with local interactions. It is in the ground state. When the system is in its ground state and there are interactions, it means that different parts of the system are entangled. They can be entangled only on short distances or longer distances, but they are entangled. When we have a pure state like the ground state, and we break the system into two subsystems, one can define so-called entanglement entropy between these two subsystems. By definition, it's the entropy of either part. If we discard the other part, the remaining part will have some nonzero entropy, von Neumann entropy.

Usually, when we have a system with local interactions, and there is an energy gap between the ground state and excited states, then correlation and entanglement are short-ranged, and the amount of entanglement is proportional to the length of the boundary of the cut region. That's called area law. The term area comes from higher dimensions. For example, when we cut a ball from a three-dimensional system, its boundary has some area. Our work was about two-dimensional systems. The entanglement is proportional to the length usually. But when the system has anyons, the entanglement is slightly less than the length. Actually, if we allowed anyons to be in the disc that we cut, it would match that simple formula. But we constrain the system not to have anyons in the disc, so there's a little bit less entanglement. And we found a formula for that difference. It's a beautiful formula in terms of so-called quantum dimensions of those anyons.

**ZIERLER:** What is an interferometric experiment?

**KITAEV:** Interferometric can mean many things. In interferometry, there's usually one particle that splits in a quantum mechanical way, meaning it forms a superposition of a particle traveling along one path and a particle traveling along another path. Then, these two particles meet. When they meet, depending on the accumulated phase on each path, the signal can be stronger or weaker. Interferometry is most commonly observed with photons. We don't have to think about individual photons, we can think about electromagnetic waves or just waves. In optics, interferometry is described by classical equations. We know there are patterns with fringes when light can go through two holes or slits. When it arrives at the screen at some distance, the signals kind of interact with each other. It's not a real interaction. The signals either add or subtract, forming a stripe pattern. That is interferometry in optics. People also do interferometry with electrons. You're referring to the two papers with Dmitri Feldman, right?

**ZIERLER:** Yes.

**KITAEV:** In those papers, we studied interferometry with quantum Hall particles. In the quantum Hall effect, there are particles, and when the effect is fractional, those particles are anyons. As the two particles meet, their relative phase depends on the presence or absence of anyons between these two paths. We were not the first to consider this problem. There had been many papers on this subject. We considered a particular type of interferometry and did the calculation.

**ZIERLER:** Looking at your work with Dorit Aharonov and John Preskill, when did you start thinking about fault-tolerant quantum computation?

**KITAEV:** I started to think about it very early, shortly after I'd started thinking about quantum computation in general because this was clearly necessary. It was necessary to find a way to protect quantum information. The first step is quantum codes. But the next step is to correct errors while doing computation, not just correcting errors in quantum memory. Fault-tolerant computation is a way to do computation when there are faults or errors, and we need to deal with them and still get the correct result. I started thinking about it, say, in 1996, and I had some ideas based on anyons, but then Shor's paper came out in 1996, and he basically solved the problem in a different way. I later wrote a paper about quantum computation with anyons in 1997, and it was motivated by the need for fault-tolerance. This was a different fault tolerance scheme. So-called topological quantum computation is just a particular form of fault-tolerant computation. With Dorit Aharonov, we wrote a paper on some technical question. I don't clearly remember that paper, but previously, she and Ben-Or came up with one particular scheme of fault-tolerant quantum computation. They proved the threshold theorem for fault-tolerant computation. It was one of several proofs that [were developed] in the late 90s. That was a significant step. We did some joint work, but somehow, I don't remember clearly what we did. It wasn't as significant as their previous work.

**ZIERLER:** And long-range correlated noise with fault-tolerant quantum computation, what does that mean?

**KITAEV:** Long-range correlated noise means that there are errors that simultaneously affect multiple qubits separated by a large distance. Usually, we assume that when an error happens, only one qubit is affected. Maybe two qubits that are adjacent to each other. But with correlated noise, many qubits can be affected, or two qubits separated by a large distance.

**ZIERLER:** Yet another question on universality, your work with John on topological entanglement entropy, what does that mean?

**KITAEV:** It's entanglement between a disc and the remaining part of the system.

**ZIERLER:** And how does that relate to broader concepts of entropy in physics and mechanics?

**KITAEV:** Entropy is a general concept. There is a quantity defined for a probability distribution and called Shannon entropy. It's also defined for a mixed quantum state; that's called von Neumann entropy. These are mathematical concepts, and there's an interesting theory describing entropy in general and properties like subadditivity, other mathematical properties. Now, entropy has applications in physics and information theory. Shannon used entropy to describe bounds on information transmission. But in physics, in particular, entropy is usually used in thermodynamics. It's a thermodynamic property of a system. These days, people compute entanglement entropy, which doesn't have such a clear physical interpretation as thermodynamic entropy. Entanglement entropy arises when we consider a pure state, then artificially divide the system into two parts, and compute the entropy of one part.

So far, it has been just a mathematical curiosity. I'm not sure if it will ever become a significant physical concept. In some cases, it matches the thermodynamic entropy, in particular when we talk about the so-called thermofield double state. The thermofield double state is related to the thermodynamic state, which is a mixed state. We can double the system and construct a pure state such that its restriction to one part gives the original mixed state. And this is, again, an artificial construction. There are models where the thermofield double has a natural physical realization. But in most cases, it's just a mathematical curiosity.

**ZIERLER:** Your proposal on a protected qubit based on a superconducting current mirror, what kind of a mirror is that?

**KITAEV:** Current mirror means it's a device such that if we pass current between two terminals, we induce current going through two other terminals. Of course, we should allow current coming in and going out, otherwise the device would break or not function properly. But if we short-circuit the other two terminals and pass current between terminals one and two, we'll drive the equal amount of current between the two other terminals in the closed loop. Such devices exist. Practical devices are classical in the sense that they don't preserve quantum coherence. You cannot use them in a quantum regime to construct superpositions between states with different currents and things like that. They won't work in this regime. But I proposed a device that could work in the quantum regime, and it could be useful to implement qubits.

**ZIERLER:** What is an electronic Mach-Zehnder interferometer?

**KITAEV:** It is the particular type of interferometer I studied with Dima Feldman. It's hard to explain. A picture is required. It is an electronic interferometer that operates in the quantum Hall regime, –let me describe a simpler type of interferometer, not Mach-Zehnder. We can have a strip of quantum Hall liquid with two constructions. There are waves, or you may say quantum particles, edge modes propagating along the edges, but in the constriction, a particle can tunnel between the two edges. If there are two constrictions, that particle can tunnel along two different paths, through one constriction or the other constriction. It's like a river, and there's some narrow place, and a particle can jump across the river. If there are two such places, it can jump in either place kind of simultaneously, meaning the particle going one way is in a quantum superposition with the particle going the other way. Now, a Mach-Zehnder interferometer is a different topology, and it requires an overpass. The strip forks, and one of these two strips should go over the other. It's hard to explain in words.

**ZIERLER:** I'm not sure how much of a role you had in this multi-author paper, it's a very interesting title, *Interacting Anyons in Topological Quantum Liquids: The Golden Chain*. What is the golden chain?

**KITAEV:** The golden chain refers to the golden ratio, it's just a number, one plus square root of five over two. This golden ratio was given some significance by the ancient Greeks. But it's just a number that often appears in mathematical problems. This chain was made of anyons, and the fusion and braiding rules for those anyons contain the golden ratio, the formulas contain the golden ratio.

**ZIERLER:** What is golden about the golden ratio?

**KITAEV:** I don't remember who proposed this, but it goes back to the very early history of mathematics. But it's not even mathematics proper because this golden ratio is kind of a perfect proportion. One can find this ratio in the human body and other objects. It's not the height and width of a person, it relates to some other things I don't remember. But it's, like, 1.6, and it looks pleasing in many cases.

**ZIERLER:** It's elegant, you're saying.

**KITAEV:** Yeah.

**ZIERLER:** Many questions on the paper *Noncommuting Flux Sectors in a Tabletop Experiment*. First, I wonder if you can talk about the use of Josephson junctions, which I know have a very rich history in physics.

**KITAEV:** We actually used the current mirror, which was made of Josephson junctions. That paper is something quite abstract. It's related to previous work by Greg Moore. In his previous work, he found that the Maxwell theory of electrodynamics–of course, it must be quantum electrodynamics–has nontrivial topological features if we put it on nontrivial manifolds, not just the three-dimensional space, but some nontrivial manifold like the three-dimensional projective space. Of course, in practice, we cannot make such manifolds because we live in flat space. We cannot curve this space and make this nontrivial topology. But in this paper, we tried to simulate nontrivial topology using those current mirrors.

**ZIERLER:** You say the paper is very abstract, and yet, in the title, it says it's a tabletop experiment. Was there an actual experiment conducted?

**KITAEV:** I don't think people will make such an experiment, but it sounds like a tabletop experiment to a field theorist because the previous work by Greg Moore was about nontrivial topologies, which are hard to come by. And compared to that, using the current mirror is easy. But for an actual experimentalist who is tasked with this problem, it will be very hard. Basically, it's something that can be done in principle in our three-dimensional world. It doesn't say it's easy to do.

**ZIERLER:** Around this time, 2006, 2007, even if this particular experiment would've been very difficult to do, had the field progressed to a point where experimentalists and theorists were working more together in a way that ultimately would lead to the creation of IQIM, adding matter to the IQI? In other words, in the very early years of quantum information, it was only theorists thinking about very abstract concepts. Then, as we get closer to 2010, 2011, more condensed-matter theorists and experimentalists become involved, and that's one of the deeper stories of the creation of IQIM. When you're thinking about Josephson junctions and superconductivity on a theoretical experiment, does that suggest that at this point in the field, there was a certain maturity that allowed for collaborations that might not have been possible a decade earlier?

**KITAEV:** There's certainly more maturity now than there was before. It's not completely true that there were only theorists in the early days of quantum computation. There were also people doing quantum optics. Quantum optics has been an established field for a few decades.

**ZIERLER:** And you would consider quantum optics to be a part of quantum information?

**KITAEV:** Well, quantum information is kind of a meta subject. I would say in the late 90s, early 2000s, new concepts were invented that belonged to quantum information proper. People figured how to do quantum computation in principle. This is very important, and this is a separate development from any concrete field in physics. These days, the subject requires practical application of these ideas, and therefore, there will be separate developments in different subfields, like doing some experiments using quantum optics, trapped atoms or ions, or doing experiments using Josephson junctions, and other physical systems. In quantum information as a theoretical subject, still, new developments happen, and some new developments are related to quantum gravity. People find connections between quantum gravity and quantum information. But on the practical side, when it comes to implementation, it's system-dependent. Quantum optics is one area where people don't necessarily focus on information-theoretic applications, they just do some very fine experiments with atoms and light. Some of them are useful to do quantum computation or other information-processing tasks. Others may just be interesting on their own. The same is true for condensed-matter systems like Josephson junctions. People can build superconducting qubits out of Josephson junctions, or they can do other things. On the experimental side, it's not that much information as concrete subjects. You need to solve concrete problems in those fields.

**ZIERLER:** I'm looking at your work on Kondo impurities, *Quasiparticle Poisoning and Josephson Current Fluctuations Induced by Kondo Impurities*. Are Kondo impurities related to the Kondo effect?

**KITAEV:** Kondo impurities certainly are related to the Kondo effect. This is a type of impurity for which Kondo wrote his model. The Kondo effect has to do with spin impurities in metals. When a localized spin interacts with electrons in the metal, there are various effects. The original Kondo effect is the increase in scattering, and therefore, an increase in electric resistivity at low temperatures due to those interactions. That paper was motivated by some idea of noise source in a superconducting device. We hypothesized that Kondo impurities cause charge noise and fluctuations of Josephson current in superconducting devices. We just came up with a model of a Kondo impurity in a superconductor.

**ZIERLER:** What is quasi-particle poisoning? What does that mean?

**KITAEV:** That refers to a process in a superconducting qubit that destroys the encoded state. To make superconducting qubits, the system needs to be at a very low temperature. The temperature needs to be low for several reasons. One of them is to suppress quasiparticles in the superconductor. At temperatures on the order of the transition temperature, there are many quasi-particles, and they will interfere with the operation of the qubit, so we want to get rid of them. For the qubit to work, the temperature should be much lower than the superconducting transition temperature. Like in aluminum, the transition occurs at about one kelvin, and the system can work, say, at 20 millikelvin, 50 times below the transition temperature. Then, the density of equilibrium quasiparticles is negligible. We should not see a quasiparticle during the whole time of experiment. If you estimate the density of those quasiparticles, it's extremely low. But in reality, there are quasiparticles, which are not at thermal equilibrium. They're produced by other mechanisms. One mechanism people are very concerned about right now is cosmic rays.

**ZIERLER:** How come?

**KITAEV:** It's very difficult to shield the system from cosmic rays. There are other mechanisms, like if the system lives in a cryostat, and some electromagnetic radiation can leak into the cryostat through various holes or imperfections in the design, like long wires that go into the cryostat, that also happens. A photon can get in and produce quasi-particles. But that can be mitigated by better design. With cosmic rays, it's hard to avoid them. They're everywhere. They're not very strong, in the sense that a cosmic particle will hit the device only once in a few seconds, but when it happens, it produces lots of quasiparticles. They basically shut down the device for some time. Those quasiparticles enter Josephson junctions and produce various errors. The device just doesn't function properly while those quasiparticles exist. Eventually, they mostly dissipate. Some of them get trapped such that they don't cause too much trouble. When they move around in the device, superconducting qubits cannot work.

**ZIERLER:** Are quasiparticles related at all to quasicrystals?

**KITAEV:** No, not at all.

**ZIERLER:** Totally separate.

**KITAEV:** Right.

**ZIERLER:** As we discussed earlier, you were involved in quasicrystal research?

**KITAEV:** Yes, I was involved in it very early. That was the first topic I worked on as a student.

**ZIERLER:** Did you follow Paul Steinhardt's quest to find a quasicrystal?

**KITAEV:** Oh, find a natural quasicrystal? Yeah, I heard his talk, it's very funny. Very nice story. I enjoyed it a lot. [Laugh]

**ZIERLER:** He's kind of an Indiana Jones as a physicist, searching for this. [Laugh] Tell me about your work with William Webb on wave function separation and resampling. What does resampling mean using a quantum computer?

**KITAEV:** It's difficult to describe exactly. It's a quantum algorithm. It's one of those algorithms that deal with real numbers. The goal is to represent real numbers in a quantum computer. You should not try to represent a particular real number because it's impossible, even with a classical computer. You always have some number of precision bits, and you can only represent a number to that precision. In a quantum computer, it would be wrong to just represent real numbers by basis states as one does in the classical computer. The correct way to represent an arbitrary number is to create a superposition of nearby numbers that differ in the last few digits of the binary representation. Resampling is a particular trick to deal with such superpositions.

### New Topological Phases

**ZIERLER:** Your paper in 2009, single author, *Periodic Table for Topological Insulators and Superconductors*. Is that to say that topological insulators and superconductors need their own periodic table?

**KITAEV:** At that time, people were discovering new topological phases, and I observed that if you restrict attention to a certain class of systems, systems with non-interacting fermions, then one can organize different phases into a table, and it has periodicity. This kind of periodicity had been known in mathematics for a number of years. It's called Bott periodicity in K theory. It was discovered by Bott, I think in 1959, as a mathematical fact. I made a connection between K theory and topological phases. This title was just something I came up with because it sounds interesting. [Laugh] But indeed, phases with different symmetries and different dimensions exhibit some periodicity. If we go one dimension higher, say from two to three dimensions, you have to change symmetry in a certain way, and you basically get the same set of phases as in two dimensions.

**ZIERLER:** That same year, in 2008, you were named a MacArthur fellow. What was it like when you received that news, and what did the MacArthur fellowship allow you to do?

**KITAEV:** Well, it felt great. I'm grateful to the MacArthur Foundation for giving me this prize. They actually allowed me to do anything with this money. It was just personal money. The goal is for people to be more flexible in doing things and to be able to spend money on things they would otherwise try to save on, like buying books in my case. In my case, it didn't mean very unusual things, but for somebody, it might mean traveling to an unusual place that might inspire some new thoughts, things like that. But for me, it was more mundane. But it's still great to have some extra money.

**ZIERLER:** I wonder, 10 or 12 years after you started thinking about these things when the field was so young, what did it mean for quantum information more generally that a prestigious organization like the MacArthur Foundation was recognizing your research?

**KITAEV:** Of course, they know better, but I guess it was just the sum of my individual results. I did some good work on quantum computation, and they recognized that work. It was, if I remember correctly, for topological quantum computation. Sorry, I didn't answer your question; I am not sure how to answer it.

**ZIERLER:** Back to the research, topological classification of free fermion systems. What are free fermion systems?

**KITAEV:** A free fermion system is a system where interactions between electrons or other fermions can be neglected. Such systems are mathematically described by a Hamiltonian that is quadratic in the creation and annihilation operators. It contains terms like c-dagger-c, and also c-dagger-c-dagger and c-c in the case of superconductors.

**ZIERLER:** Your work, *Topology-Driven Quantum Phase Transitions in Time-Reversal-Invariant Anyonic Quantum Liquids.* Are these actual liquids or a theoretical construct?

**KITAEV:** It's a theoretical construct. People talk about electron liquids and spin liquids, and the meaning of the word liquid is probably not what you think. It doesn't mean the system is actually fluid. It means it's different from, say, a crystal or a gas. Let me be more specific. An ordinary material like water can be in a solid phase, where atoms or molecules are arranged periodically. It can be in a liquid phase, and it can be in a gas phase. Solids are characterized by order. It's hard to explain all these concepts, but intuitively, it means that the molecules are arranged periodically. In particular, they are oriented in certain direction. This orientation is preferred by the system compared to any other orientation. Also, certain positions are preferred to other positions because atoms are arranged periodically. In a liquid or a gas, there is no order. In fact, liquid and gas are not fundamentally different phases. These are just two extreme cases of one phase. But it's important that in a gas, interactions are weak because the density is low. In a liquid, molecules are tightly packed, and the interactions are strong.

By analogy, people distinguish solids in spin systems where there is an ordering. For example, an anti-ferromagnet is a simple example of a solid, so to speak, where spins are periodically arranged. The spins themselves are locked in the crystal lattice. They cannot move, but they can rotate. In an anti-ferromagnet, they form a checkerboard pattern if it's a two-dimensional system. Spin-up and spin-down are like the white and black squares in a checkerboard pattern. There are other solids where spins are arranged differently. They might not have a preferred direction, they might break into pairs, and each pair will be entangled between themselves, but there will be separate entangled pairs. A liquid is something that doesn't have order at all. Usually, those liquids have other interesting properties instead of ordering. They might harbor anyons. The liquid is the ground state, and quasiparticles or excitations in the system are anyons. Instead of having order, they have other interesting properties, more interesting than order.

**ZIERLER:** Many questions on the paper, *Power-Law Spin Correlations in a Perturbed Honeycomb Spin Model*. Let's start with the honeycomb spin model. What is that?

**KITAEV:** The honeycomb spin model is that model I proposed in the paper, anyons in an exactly solvable model and beyond. But that's the model.

**ZIERLER:** This is in 2010 with Tikhonov and Feigel'man.

**KITAEV:** I vaguely remember because my contribution was negligible on this paper.

**ZIERLER:** What are power-law spin correlations?

**KITAEV:** As a function of the distance between the two spins, it decays as a power law. One over the distance squared, for example. That should be contrasted with exponential decay. Like E to the minus the distance.

**ZIERLER:** That same year, you looked at topological phases of fermions in one dimension. Are there other dimensions that these phases exist in?

**KITAEV:** Topological phases of fermions exist in all dimensions. But we studied a concrete model in one dimension with Lukasz Fidkowski.

**ZIERLER:** What is that one dimension? What does that mean? If they're in all dimensions, what's the focus on one dimension?

**KITAEV:** It's a chain of fermionic sites. There are sites between which fermions can move. In one dimension, things are easier than in two dimensions and even easier than in three dimensions. We considered a concrete question in one dimension and solved it. But it's very hard to explain it because it's a technical question.

**ZIERLER:** But are they the dimensions that we think about that govern time and space?

**KITAEV:** They govern space. Of course, we live in there spatial dimensions and one time dimension. Mathematically, we can consider problems in any number of spatial dimensions. It's not clear how to deal with two-dimensional time. People don't do that usually. But one can add more spatial dimensions. In practice, all dimensions from zero to three can be realized. Zero means it's just a very small system that consists maybe of a few atoms that interact somehow. One means that it's in a chain of atoms. There are crystals that consist of individual chains. Each chain works as a one-dimensional system. There are also engineered systems like nanowires. Then, you have just a single wire where electrons move strictly in one dimension. They can wiggle a bit in the perpendicular direction, but at low temperature, this wiggling is suppressed. One can do two-dimensional, for example, quantum Hall systems. In two-dimensional systems, electrons are confined to some layer, and it can be either a layer cake of different materials, where one layer carries electrons, or it can be a two-dimensional material like graphene. Graphene is a popular material now. It's strictly two-dimensional, just one sheet of carbon atoms. This is how one- and two-dimensional systems are made. Therefore, it's practical to study some models in one and two dimensions. Hopefully, those models can be realized experimentally.

**ZIERLER:** Is graphene going to be an important material for building a scalable quantum computer?

**KITAEV:** I don't know. People are not using graphene right now. But who knows?

**ZIERLER:** What are some of the obvious materials even now that are likely to be relevant for building scalable quantum computers?

**KITAEV:** I cannot tell. Right now, people are using aluminum for superconducting quantum computers. There are actually several technologies that might produce a quantum computer eventually. Existing technologies. They include trapped atoms. It's not a material, just atoms in a vacuum that are trapped by electromagnetic fields. Another technology is based on superconductors. The most popular superconductor is aluminum, which is used not because it's a good superconductor. It's superconducting only at temperatures below one kelvin. It's used because people know how to make good Josephson junctions out of aluminum oxide. That's the only reason. It's a technological reason. They just developed the technique of making this structure. They create a very thin film of aluminum oxide and then can deposit another layer of aluminum. Only because they know how to do it, they use it. If people figure out how to do the same thing with, say, niobium–they kind of know, but it's hard. If they can scale up such Josephson junction and improve the quality, they might switch to niobium because it has higher superconducting transition temperature. I don't know.

**ZIERLER:** Is this the approach that Google is pursuing, using aluminum for superconductivity?

**KITAEV:** Yeah. Currently, I believe all groups use aluminum.

**ZIERLER:** Including Microsoft?

**KITAEV:** No, Microsoft uses something else entirely. They're trying to use nanowires made of indium arsenide. Nanowires per se might not work, physical nanowires that are grown as individual crystals, because they have a lot of defects on the surface. They're trying to improve the technology. I don't know what exactly they're doing right now, but one way is to fabricate those wires in a planar structure. It won't be physically cut wire; it will be just some potential that confines electrons to a one-dimensional region. I'm not sure if they're pursuing this path, but somehow, they need to solve the problem of impurities, which exist in large numbers in those nanowires.

**ZIERLER:** Back to the research. In 2011, your work with Liang Kong on building models for gapped boundaries and domain walls. What are gapped boundaries?

**KITAEV:** I'll start with what a gap means in general. A gap means an energy gap. If we build some material and cool it down to a very low temperature, the individual degrees of freedom in that material, like electrons, form a ground state. Producing a particle may be easy or hard. It may require a finite amount of energy. Suppose the system is at five millikelvin. Aluminum, as we previously discussed, carries Bogolyubov quasiparticles, and their energy is on the order of two kelvin. That is the energy gap. One needs to put a certain amount of energy to produce an excitation. Below that, no excitations will be produced. If it's not aluminum but copper, it's a gapless system. Copper remains non-superconducting at those temperatures. One can produce an excitation, like an electron or a hole, with arbitrary small energies. That is called gapless. Now, certain systems are gapped in the bulk, but their boundary is gapless. Three-dimensional materials like that are called topological insulators or topological superconductors. Particularly in a topological insulator, there is an energy gap in the bulk, but electrons at the surface do not require any gap, any fixed energy to be added or removed. You talked about the paper on topological systems. This was a class of two-dimensional topological systems that are gapped both in the bulk and on the boundary.

**ZIERLER:** Finally, last question for today, same year in 2011. Tell me about developing a technique on dissipation-less dynamics of randomly coupled spins at high temperatures. What does that mean, dissipation-less dynamics?

**KITAEV:** I've looked at the paper, and now I understand why it's called dissipation-less dynamics. We considered the model of interacting spins. This dissipation-less word has to do with the justification of the model. We had to justify that those spins only interact between themselves and nothing else. Typically, spins interact with other degrees of freedom in the system, which causes dissipation of energy. Basically, the energy can exchange between the spin subsystem and other systems. We considered spins embedded in a superconductor, where there is an energy gap, as we discussed previously. When spins flip, they cannot excite quasi-particles in the superconductor. That's why the only important interaction is the interaction between the spins themselves. We solved a particular problem of their dynamics, calculated fluctuations that occur at various time scales, both long and short.

**ZIERLER:** Why the focus on high temperatures? What does that mean here?

**KITAEV:** High temperature means that the temperature is greater than the interaction energy between two spins. And that energy is extremely low. Temperature like 50 millikelvin would be considered as high for this purpose.

**ZIERLER:** If we can translate millikelvin into Fahrenheit or Celsius, what temperature are we looking at?

**KITAEV:** The absolute zero is -273 Celsius. Above that, it will be just 1/20 of a Celsius.

**ZIERLER:** So when you mean high temperatures, they're actually quite cold in the way that we perceive temperatures.

**KITAEV:** Yeah, quite cold for us but common for a condensed matter experiment. Physicists work with absolute temperatures. Temperatures above that specific temperature, which corresponds to absolute zero and means no thermal energy.

**ZIERLER:** A very different conception of temperatures, of course.

**KITAEV:** Yeah.

**ZIERLER:** Well, Alexei, on that note, we'll pick up next time from 2012 and bring the story right up to the present.

[End of Recording]

**ZIERLER:** This is David Zierler, Director of the Caltech Heritage Project. It is Thursday, February 10, 2022. I'm delighted to be back with Professor Alexei Kitaev. Alexei, it is good to be with you again.

**KITAEV:** Thank you.

**ZIERLER:** Today, I'd like to start in the year 2012 with something that you might consider an interesting question. You were part of the inaugural class of Breakthrough Prize Winners. First, did you ever know Yuri Milner before the prize was announced?

**KITAEV:** No.

**ZIERLER:** Never met him before?

**KITAEV:** No.

### Testing Quantum Information

**ZIERLER:** Here's where I'd be very interested to hear what you have to say about this. If you look at the list of people you shared the Breakthrough Prize with, first, there's Alan Guth, and there, it's cosmic inflation, something that there are very exciting possibilities about and experimental or observational demonstration that the theory of inflation is correct. Andrei Linde, of course, about the multiverse and what he believes is a testable proposition, that the multiverse is something that we could put to the rigors of science. Then, when you look at Nima Arkhani-Hamed, Maxim Kontsevich, Juan Maldacena, Ashoke Sen, Nati Seiberg, particularly Ed Witten, all of them are related to string theory, and of course, there are things in string theory that are waiting to be experimentally validated as well. I wonder where you see quantum information in this regard. In other words, to the extent that the Breakthrough Prize awards excellence in physics, even though there might not be the testable demonstrations that, for example, the Nobel Prize committee is waiting for when they award a Nobel Prize. What are your thoughts on that?

**KITAEV:** Regarding testable demonstrations, I think quantum information is easier, although there hasn't been a testable demonstration of a quantum computer that would solve some real problems. By now, there has already been a demonstration of basic principles. There's no reason to doubt quantum mechanics, which was discovered almost 100 years ago. To me, there's very little doubt about the possibility of a quantum computer. It's just a difficult experimental and technological task. The difference is, quantum information or quantum computing is kind of an applied science. The other work you mentioned like string theory is more fundamental. That's the main difference. On the other hand, the novelty of quantum information science was that before, say, the 1980s, people didn't try to think about quantum mechanics in a complex setting. They thought there were natural processes, the laws of nature could not be changed, and the systems they studied were also given. The problem was to infer some parameters of that system from experimental data. Quantum information is about making complex quantum systems and controlling them, and they may exhibit some new phenomena that were not possible in natural systems. This is a novel aspect. It first appeared before my work, it appeared in the work by Bennett, Feynman, and others. Peter Shor created the foundation of quantum computing by showing how to correct errors. That enables computation on a large scale in principle.

**ZIERLER:** You mentioned the expectation, or there being a benchmark, that we need to have a quantum computer that has some social utility or usefulness. To what extent is that a fair expectation, certainly in light of the fact that, depending on how you define it, we do already have quantum computers, even if on a very small and not particularly useful scale today?

**KITAEV:** I think the expectation comes from the fact that people are not exactly interested in quantum computing, they're interested in things that can be done with the quantum computer. This is very natural. It's not about the demonstration of principles, people just want to see it working, and that's it. [Laugh] I also share this expectation because, as I said, to me, there's no doubt that the foundations are solid, and if everything is done correctly, it will work. But I still want to see it actually work.

**ZIERLER:** For your own research, more broadly defined in quantum information generally, do you see any of the research theoretically that has been experimentally or observationally demonstrated?

**KITAEV:** Surface codes are a good example. They've been constructed by the Google group and have been demonstrated to work. But on the other hand, it's a mathematical construction, so there's nothing that can possibly fail in the construction itself. It's more like an application of this construction in practice, and I'm very happy that it happened.

**ZIERLER:** We'll return to some nontechnical questions and cover some of your key papers from the last ten years. In 2013, the paper *An Area Law and Sub-Exponential Algorithm for 1D Systems*. First, what is a 1D system?

**KITAEV:** A one-dimensional system. In this case, a one-dimensional Hamiltonian. There are some units like spins or other simple systems with a small number of quantum states, and they interact with their neighbors. They're arranged in a sequence, in a chain, and the nearest neighbors interact.

**ZIERLER:** As you note with your coauthors on this paper, it improved exponentially the results from Hastings. What was Hastings's result, and how did this work improve upon it?

**KITAEV:** It's a mathematical physics result. It's a statement about ground states of certain Hamiltonians. As I said, the Hamiltonian involves interactions between neighbors. There's another technical condition, the existence of an energy gap, meaning that excited states have some smallest energy. One cannot create an excitation with arbitrarily small energy relative to the ground state. Hastings considered cutting the system in half and calculating the entropy of one half, which is called entanglement entropy because the whole system is in a pure state and its entropy is zero. But if we consider half of this system, its entropy is finite. Hastings obtained a bound on this entanglement entropy in terms of the energy gap. He proved that the entropy cannot be greater than some function of the energy gap divided by the interaction energy. It is also a function of the number of states in each subsystem, in each block of which this chain is built. We obtained a better bound.

**ZIERLER:** Back at Caltech, were you involved in the creation of the IQIM, the transition from the IQI to the IQIM?

**KITAEV:** No, I wasn't involved in any organizational things at all. I recall how the transition occurred. There was, of course, a proposal, and I participated a little bit in writing it, but most of the work was done by other people. I wrote a few paragraphs regarding what I was going to do.

**ZIERLER:** Perhaps in your discussions with John or more generally your appreciation of what was going on at the time, why this time period, 2011 to 2013? What was going on in the field of quantum information that suggested the time was right to add matter to the IQI? What did that mean for the field?

**KITAEV:** Of course, this is useful for the field because we're interested in doing complex quantum experiments with various systems, and I mentioned last time that quantum optics and cold atoms were the first systems on which complex quantum experiments were done. But we're also interested in doing this in condensed-matter systems. People naturally had started that before IQIM, but getting more funds for such work is important.

**ZIERLER:** In 2013, with Peter Brooks and John Preskill, you wrote a paper *Protected Gates for Superconducting Qubits*. What are protected gates, and are the gates doing the protecting, or are they being protected?

**KITAEV:** They are protected in the sense that they're designed in a certain way such that errors that might occur during the implementation of the gate do not affect the result very much. In this sense, they are protected.

**ZIERLER:** The focus on superconducting circuits, do you see from this work a particular connection with certain industries that have approached quantum computers from a superconducting circuit perspective?

**KITAEV:** Superconducting circuits are one technology. It's an alternative to cold atoms and ion technology. Those circuits are built of elements like Josephson junctions, which is the most important element. Right now, Google is building a quantum computer using superconducting circuits. It's good that they have enough resources to build large enough systems and do complex things. But people in various labs also do that on a smaller scale.

**ZIERLER:** Do you see superconducting circuits as the most plausible approach to building a scalable quantum computer?

**KITAEV:** I don't know. I'm involved in the work of Google, so I hope superconducting circuits will be the most viable approach, but I can't tell.

**ZIERLER:** Do you have a sense of why Google adopted this approach as opposed to trapped ions?

**KITAEV:** I don't know. But the Google group was initially led by John Martinis, who specializes in superconducting circuits. I don't know, it just happened this way.

**ZIERLER:** In 2015, you won the Dirac Medal. I wonder in what ways this recognition was meaningful to you, particularly in light of Dirac's fundamental contributions to quantum mechanics, and thus, ultimately, to quantum information.

**KITAEV:** Yes, of course. I'm very honored to receive the Dirac Medal. It was nice to travel to Trieste and participate in the ceremony. The Dirac Medal is a prestigious award. I'm proud to receive it.

**ZIERLER:** We talked earlier about Richard Feynman's ideas and how they led to the field of quantum information. Do you see that going back even farther? Do you see an intellectual connection that goes as far back as Dirac, for example?

**KITAEV:** No, I don't. Regarding quantum information, Feynman was one of the first people, aside from Yuri Manin. They were the first to consider the idea that quantum mechanics could be useful for computation.

**ZIERLER:** In 2017, *Efficient Encoding for the Hayden-Preskill Protocol*, where you looked at Hawking radiation. First, what is the Hayden-Preskill protocol?

**KITAEV:** It's a bit difficult to explain. Basically, it's about recovering information from a complex system where information may be scrambled. Hayden and Preskill made a nice story to introduce the problem, and it involves black holes. The story goes that Alice wants to hide or destroy her diary containing private thoughts, and Bob is trying to spy on Alice. Alice thinks that a black hole might be the safest place to get rid of her diary. She throws it into the black hole, but Bob can, in principle, recover the information from the Hawking radiation. Indeed, there are many interesting information theoretic properties of black holes. But this particular task is not related to black holes specifically, it's about any complex system. Suppose we perturb a complex system a little bit, let's think about a piece of metal at a certain temperature, and so we do a small perturbation, maybe send a laser pulse that heats it a little bit. It's a small perturbation to the system, and any effect of that perturbation quickly dissipates. If some energy's added to the system, it is only reflected in the temperature.

The system heats up a little bit. But the actual information seems to be lost. In principle, if it's a quantum system, one can decode its quantum state and recover the information, that is, determine what exact perturbation happened. But suppose we don't have full access to the quantum state, we can only access a small piece of the system. Then, the recovery's not possible. However, Hayden and Preskill noted that if this quantum system was previously entangled with another system that we fully control, such as a quantum computer, then, using that device and a little piece of the system itself, we can recover the perturbation. Hayden and Preskill showed that the recovery is information-theoretically possible. Beni Yoshida and I just gave an algorithm. It's a quantum algorithm how to recover the information.

**ZIERLER:** Was this the first time, around 2016, 2017 with the Hayden-Preskill protocol, that you started thinking about black holes and Hawking radiation? Or is this related to earlier research at all?

**KITAEV:** It's related to research I did a little earlier. Of course, I have been always fascinated by black holes. The Hawking information paradox is one puzzle that bothered me. I knew about that a long time ago. The work by Hayden and Preskill was very exciting. Of course, it didn't solve the paradox, but it shed some light on how black holes might work. I started thinking about this problem seriously, I think, in 2014. As a result, I came up with the SYK model and maximal chaos in the SYK model. That was in 2015. But I started working on it in 2014.

**ZIERLER:** To clarify, are you thinking about black holes purely as mathematical constructs or theories, or is there something going on observationally with real black holes? For example, were you following developments with LIGO, or did you ever talk with Kip Thorne about black holes?

**KITAEV:** I have followed the developments of LIGO to some extent, and it's very exciting that people can see actual signals from black holes. The black holes I study are kind toy black holes, not much related to real black holes. My actual interest is rather theoretical and abstract. But I'm still excited about the LIGO developments in astrophysics because there are so many interesting things happening in the universe. It's very exciting to know about that.

**ZIERLER:** To clarify further, does a toy black hole mean that for the purposes of your theoretical work, it could behave in ways that are different than a real black hole?

**KITAEV:** It behaves differently because one can make it respond in a certain way, which we cannot do with a real black hole. A real black hole is big, and if you look at the formulas for the Hawking radiation, the Hawking radiation is negligible. We cannot see it in practice. Furthermore, I'm interested in more subtle properties than just Hawking radiation, like finding complex correlations in the Hawking radiation. That's completely impossible with a real black hole. I'm thinking about those toy black holes, which are just mathematical models sharing some properties with real black holes. But they're actually quite different.

**ZIERLER:** Yet another prestigious prize, in 2017, along with Xiao-Gang Wen, you were the winner of the Oliver Buckley Prize in condensed matter. First, did you work at all with Wen?

**KITAEV:** No, I've never worked with Wen.

**ZIERLER:** What does the Buckley Prize say more broadly to you about the connections between condensed-matter physics and quantum information?

**KITAEV:** I can speak to what I did and how these subjects are connected in my work. I was interested in quantum codes, and I realized that certain condensed matter systems can encode quantum information such that it is protected from errors. They effectively work as quantum error-correcting codes. These systems have nontrivial topological properties, and they carry excitations called anyons. In particular, one could try using quantum Hall systems. Fractional quantum Hall systems are known to carry anyons, although anyons are hard to observe experimentally. There are other systems with nontrivial topological properties that can also be used in principle to encode quantum information. For example, Majorana wires are very thin wires made of certain semiconductors in contact with a superconductor and in a magnetic field. They feature boundary Majorana modes. At the ends of those wires, there are what's called Majorana modes, special places that host part of a quantum state. The quantum state is actually in the whole wire, but one can access it through the end. This can be used to store a qubit. The whole wire can be used to store a qubit. This is another interesting condensed-matter system. The Buckley Prize, I received it for my work on such topological quantum systems. Also, my other awards were for this work and its connection to quantum information. I studied topological quantum systems. Wen also studied topological quantum systems. He started earlier than I. But he didn't make an explicit connection to quantum information, as far as I know. In his later work, there was connection. He and Levin wrote a paper similar to our paper with John Preskill. They derived the same formula for the topological entropy of a disk in a two-dimensional system that carries anyons.

**ZIERLER:** We've talked previously about the SYK model. In 2017, along with Josephine Suh, *The Soft Mode in the Sachdev-Ye-Kitaev Model and its Gravity Dual*. First, to understand, did you ever work directly with Sachdev and Ye? Or did you modify their work independent of them?

**KITAEV:** I modified their work independent of them.

**ZIERLER:** What were the opportunities? Where did you see need or opportunity to modify or improve upon what they had done?

**KITAEV:** The story is this. I mentioned that in 2014, I started thinking about information in black holes. I realized that one can first do something interesting about black hole horizons in connection with quantum information. Second, one can access the black hole horizon from the outside if one studies so-called out-of-time-order correlators. I was looking for a model where out-of-time-order correlators would be similar to those in black holes. But the system itself would be quite different from a black hole. I had some understanding of what kind of system it should be. I was looking through the literature, just looking for an interesting system that might be good for that purpose, and I came across the paper by Sachdev and Ye, and the system seemed promising. But it was also rather complex. I figured out how to simplify it, so I came up with a slightly different model that shares some interesting features but is easier to study. Then, I calculated the Lyapunov exponent, how fast the out-of-time-order correlators grow, and found that it's the same function as in black holes.

**ZIERLER:** The soft mode in the SYK model, what does that mean?

**KITAEV:** The soft mode is a general concept in physics. When we build a many-body system, or it can be some natural system in particle physics, there are excitations with high energy and low energy. Soft modes are basically low energy excitations that are easy to produce. Soft means it's easy to deform the system. The energy required for that is much less than the elementary interaction energy in the Hamiltonian.

**ZIERLER:** What are out-of-time-order correlators? What is out of time?

**KITAEV:** When we make an observation on a system, we first perturb the system and later measure the effect of that perturbation. Such an experiment can be mathematically described by a response function, and it's also related to a correlation function between operators. Usually, when we consider such a process and construct the corresponding correlation function, the times on the operators are going in a certain order. Usually, the times increase and then decrease. For example, 0, 1, 1, 0, something like that. There is a certain sequence of times that enters the expression; it goes forward in time, then backward in time. Essentially, the natural evolution goes forward in time, but for some technical reason, the sequence has to be doubled. An out-of-time-order correlator is where the times on the operators alternate, like zero, one, zero, one. And such correlators do not occur in any expression for a naturally observable quantity. It's related to things that can be observed if we actually reverse the direction of time, which has been done, by the way, by Google. They studied certain quantum evolutions, and then they run the same evolution backward. In the meantime, they introduce some perturbation. By running the evolution forward and backward and doing a measurement, you can see some new things, they're related to out-of-time-order correlators.

**ZIERLER:** Another collaboration with Josephine Suh the following year, *Statistical Mechanics of a Two-Dimensional Black Hole*. To clarify, is this also a toy black hole that you're thinking of, and that's why it as two dimensions?

**KITAEV:** Yeah, it's a toy black hole, exactly.

**ZIERLER:** How do you apply statistical mechanics to black holes? What does that mean?

**KITAEV:** Statistical mechanics studies properties of a system at a finite temperature. We just considered this black hole at a finite temperature. Black holes should be considered at a finite temperature, the Hawking temperature. They have an actual temperature associated with them that depends on their energy. This is not special to the toy black hole. But of course, two-dimensional black holes are special, and we considered a simple model and solved it.

**ZIERLER:** I suppose if it's a toy black hole, it can be any number of dimensions that you want. How many numbers are in a real black hole?

**KITAEV:** Three plus one. Three spatial dimensions and one time dimension. It's four dimensions. The black hole we considered was two-dimensional in this sense.

**ZIERLER:** The reduction to two particles in the anti-de Sitter space, what are the two particles?

**KITAEV:** Oh, it's quite technical. First, we consider two-sided black holes, which do not occur in nature. These are artificial constructs. They are like if we entangle two identical black holes in completely different universes. But entanglement is a quantum concept. If we think of the black holes classically, they're connected by a so-called wormhole. They share some part of space, but that region of space is usually not traversable, so one can enter it from either side but cannot go back or to the other side, except by going back in time.

**ZIERLER:** We'll take a brief break from the technical discussion for a question that historians of physics are going to be very interested in. If you look back 20 or 25 years, let's say when you were at Microsoft, and the quantum information effort was very small and of course very theoretical, then you fast forward to today, and all of these major companies, startups, worldwide effort, there's so much interest, input, and excitement about quantum information and quantum computing. Between then and now, over these 25 years, is your sense more that the increasing interest and excitement in quantum information was gradual? Or were there certain advances in the field along the way that created bursts of energy and resources that explain how we are where we are today?

**KITAEV:** To me, it looks like a gradual process. The technologies advance. On the other hand, there's more funding, more interest in quantum information these days than in the year 2000. Now, indeed, many companies are investing in quantum technologies. I don't know whether it will be successful on the time scale that is relevant to those companies. In the end, it will be successful. I'm sure about that. But I don't know how soon.

**ZIERLER:** Of course, you're not a businessperson, but to the extent that you understand why major companies, governments, and even startups are investing so much money, for people who run the finances but are not specialists themselves, what is your sense of their motivations? Why are they excited about this? What do they believe makes this such a good investment, even if the investment will pay off many years from now?

**KITAEV:** I don't know. There are companies that really care about the distant future and make long-term plans. But I guess in most cases, this is something related to their business that I don't understand. How finance works, how social effects work, things like that.

**ZIERLER:** If only passively, in other words, not in any way that you have given a lecture where you're convincing people to invest in these technologies, how do you see your research contributing to people believing that this is an investment worth making?

**KITAEV:** I don't know. Well, I suppose that when people see this result about surface codes, they become more confident in quantum technologies. Surface codes is something I worked on theoretically some time ago. In this case, it helps to make people confident in this technology.

**ZIERLER:** A question about collaboration as it relates to two recent papers, *Observation of Time Crystalline Eigenstate Order on a Quantum Processor*, and *Information Scrambling in Computationally Complex Quantum Circuits*. You are a collaborator with dozens and dozens of people on these papers. I'm curious if you see that as a trend in quantum information, that significant papers are more likely now to be a major collaboration between tens or even dozens of scholars, and single-author or dual-author papers are becoming less likely.

**KITAEV:** These papers are written by people in the Google group. They basically put the names of all the people in the group. For example, I didn't participate in the time crystal paper at all. I had some discussions regarding information scrambling, so I might say something about it, if you ask.

### Google and Qubits and Correlators

**ZIERLER:** Please do. I'd love for you to say something about it.

**KITAEV:** This is related to out-of-time-order correlators. It's an experiment that was done using the Google machine with 53 qubits. The qubits are initialized somehow, then we run a number of gates in parallel. Each pair of qubits is made to interact; then we run another set of gates, pairing up the qubits differently. These are called layers of a quantum circuit. We apply some number of layers, then we introduce a perturbation, an extra gate, then we reverse the whole process, except that perturbation. First, consider what happens when no perturbation is made. We apply those interactions and then undo them. The system should come to the same state it was in before. Say, all the qubits in state zero. But if there is a perturbation in between, then this perturbation will affect some qubits, and one can study how this perturbation spreads as a function of the number of circuit layers. This kind of work was done experimentally, but there are several variations.

For example, one can apply gates for which the system is not chaotic, and one can see that the behavior is different. One can also calculate certain things theoretically because these out-of-time-order correlators are described by relatively simple theories. The things that could not be computed theoretically were done using numerical simulation. Basically, the average values were computed theoretically, and fluctuations around the average values were computed numerically. The results were compared with the experiment. They are consistent, although the real system has a lot of noise. When those layers of the circuit are run and then reversed, the system doesn't quite come to the state it was before, it comes to a state that is correlated with the state that was before. One has to modify the theory and take this noise into account to make a correspondence between the theory and the experiment.

**ZIERLER:** On the second paper, is information scrambling related at all to cryptography?

**KITAEV:** No. I guess it could be related to cryptography, but the way we discussed it is not related to cryptography.

**ZIERLER:** As the paper notes, quantum scrambling can be very useful for physics research. What are some of the problems in physics for which quantum scrambling might be useful?

**KITAEV:** Well, it's a theoretical problem one can consider. While doing that, one might discover some interesting properties of concrete physical systems. I think it's useful in this sense. I mentioned the SYK model. My logic was reverse. I was looking for a system with certain out-of-time-order correlators, and I found one, but one can see how it could happen the other way around if one studied a system like the SYK model. There are, in fact, related models. There's the so-called Maldacena-Qi model that consists of two copies of the SYK model coupled to each other. There, some interesting properties are related to ordinary correlators one could study experimentally when one is able to make the system experimentally. But these properties are also related to out-of-time-order correlators in the SYK model.

**ZIERLER:** More recently, a very exciting announcement at Caltech. Dr. Allen and Charlotte Ginsburg are supporting a new quantum center here. Were you involved at all in that process?

**KITAEV:** No, I'm not involved.

**ZIERLER:** Do you see any overlap with the IQIM or any possible areas of joint work with this new quantum center?

**KITAEV:** Yes, sure. It's related to the same types of problems, so there should be coordination of activities.

**ZIERLER:** It begs the question, why the need for a new quantum center? Are there ambitions with the center that are simply too large for IQIM to house itself?

**KITAEV:** I think so. Although I am not involved in this, my general understanding is that this new center will have a new building and new labs. The IQIM funding is not enough for that, it should be something on a bigger scale. If it happens on this scale, it will be a totally new level of activity.

**ZIERLER:** Do you see advances potentially there in labs experimentally that will be particularly relevant for your research?

**KITAEV:** I'm a theorist, so I think I'll just keep doing what I'm doing. But of course, I'll be glad to talk with people. Maybe at some point, I will be able to help them with some advice. As for myself, I'm a theorist, and I'm focusing on what I can do well.

**ZIERLER:** On the question of you keeping on doing what you're doing, regarding your most recent paper, *Two-Way Approach to Out-of-Time-Order Correlators*, what is some of your current work on out-of-time-order correlators, and how does this relate to the SYK model?

**KITAEV:** We studied models like the SYK model. It's a class of model where small elements interact with each other, like everybody is connected to everybody. In this class of models, we just got a better understanding of out-of-time-order correlators. Again, it's a technical result. Two ways means that we solved some equations not forward in time or backward in time, we've solved two sets of equations in opposite directions, and then obtained the final answer by combining these two solutions.

**ZIERLER:** Also, in 2021, you were elected to the National Academy of Science. What was that like, and what do you think that meant more broadly for the field of quantum information?

**KITAEV:** I don't know what it means for the field. For myself, I'm honored to be elected to the National Academy of Science. It's a recognition of my work.

**ZIERLER:** Of course, this has all happened during the pandemic. Are there opportunities to travel to conferences, to meet other people in the Academy that you're looking forward to doing?

**KITAEV:** I'll probably travel this year to Washington to participate in a meeting. But people travel these days. I've been to several conferences.

**ZIERLER:** Now that we've worked up to the present, I'd like to end our talk with some general retrospective questions, then we'll end looking to the future. One thing we haven't covered yet is your work as a professor and educator. Do you have opportunity at Caltech to interface with undergraduates at all?

**KITAEV:** Right now, I'm teaching quantum computation, so I interact with undergraduates when teaching some courses. I don't interact very much outside of this.

**ZIERLER:** As you well know at Caltech, computer science is now the dominant major among undergraduates. Are the students interested in quantum computation generally interested in computer science? Or are these more students who have a physics interest, and that's why they're coming to you?

**KITAEV:** There are both types of people. There are more physics students than CS students, but some CS students are also interested.

**ZIERLER:** If there are undergraduates thinking about their careers and looking for advice, are there jobs in quantum computation and quantum information on the horizon in the way that, for example, there are many jobs in computer science?

**KITAEV:** I cannot give any advice for computer science students. For physics students, yes, there are jobs that involve building quantum computers, like I said, at Google, Microsoft, IBM, and smaller companies. With this, it's important to know both physics and some computer science. It's an exciting field right now. I would also give a warning that it might not be funded, say, 15 years from now.

**ZIERLER:** Right. We'll talk about that. It's an interesting question. For graduate students and even post-docs that you've worked with, just generally over the years, what has been the value to you, for your research, in mentoring graduate students and post-docs?

**KITAEV:** First of all, I am not very good at interacting with students. I have only had a few graduate students in my career. Of course, for them, the value is just learning physics, learning how to work in the field. I hope my mentoring helped them. Other than that, I don't know. I'm a bit ashamed I can't do a better job, but I guess it's my personality, I'm not very good at interacting with people.

**ZIERLER:** We've touched on it, but a more direct question. If you look at all of your contributions in quantum information, quantum computation, are there specific aspects of your work that you see ultimately as being most relevant to the creation of a scalable quantum computer?

**KITAEV:** I mentioned surface codes. I think this is one piece that's practically useful.

**ZIERLER:** What is it about surface codes that you elevate above all the other things you've done?

**KITAEV:** I see it implemented in hardware. It's pretty easy to implement compared to other ideas. One can think about a surface code as a code, so there are qubits, and there are some check operators, which one can measure. In this way, one can detect errors. Those qubits can be put on a plane such that those check operators are local. One needs to measure a combination of four qubits and get a bit, zero or one. If it's a one, it means that there's an error in one of those qubits. We don't know which one, but by combining such measurements, one can figure out which qubit's in error. This is the rough idea. It's a special type of an error-correcting code that is relatively easy to implement. I had other ideas that might also work but have not worked so far. In particular, the Majorana chain.

I wrote a theoretical paper in 2000 where I put forward the basic concept, a one-dimensional system with Majorana modes at the ends that has special interactions between the elementary block of that system. Later, around 2010, a few people, including Gil Refael at Caltech, figured how to do it in a real material using a semiconductor wire with strong spin orbit interaction in connection with a superconductor and inner magnetic field. There has been a lot of work funded in part by Microsoft, including the work of Leo Kouwenhoven, on realizing those Majorana modes. The first reports were in 2012. Unfortunately, it's still inconclusive. They've seen some signatures of the Majorana modes in the experiments, but the system is not clean enough such that one can tell for sure they're Majorana modes. It could be something else, some effect of disorder. Making the system cleaner is a hard experimental problem.

**ZIERLER:** Perhaps this is a more tangible question. What about the impact of your research on physics itself? In what ways has your research advanced problems in physics that were much more difficult or did not seem easily solvable prior to the origins of quantum information?

**KITAEV:** Throughout my career, I've studied some models. I like simple models, which may be called toy models. I've found and studied several such models. This surface code is related to a Hamiltonian. One can write the Hamiltonian corresponding to the surface code. The code that was implemented at Google is not a Hamiltonian, it's more like the computer science element in this idea that is used right now. But regarding the physics element, there is this Toric code Hamiltonian that has not been implemented yet. There's another model on the honeycomb lattice, and people found some experimental realization, although it may not be exactly the same as in theory, but hopefully it's related. I don't think I can help by making some hard problems easier, I just have found some new types of problems and solved them.

### Imagining the Benefit of Quantum Computers

**ZIERLER:** Regardless of the commercial or social value in the future for quantum computers, maybe we'll get there, maybe we won't, but what are you most excited about for physics itself? What are some of the things quantum computers might help resolve that currently are not feasible?

**KITAEV:** That's a very good question. Quantum computers might help simulate complex physical systems. One problem is high-Tc superconductors, which were discovered in 1986, if I remember correctly, a long time ago.

**ZIERLER:** The Woodstock Conference at APS!

**KITAEV:** It's very exciting, but we still don't quite understand why they have these properties. One reason is, it's hard to make a well-controlled experiment. We have real systems, which include disorder, and interactions cannot be controlled easily, and not all quantities can be measured. That's why it's hard to extract the immediate data from the experiment. People also do simulations on classical computers, but of course, one cannot simulate a complex quantum system like that. One has to do some simplifications, and it's not clear whether those simplifications alter the results or capture the essential properties. Doing a simulation on a quantum computer could help resolve all these issue and find which theories are correct. There are several theories one of which is probably correct.

**ZIERLER:** Maybe it's as much a philosophical question as a scientific question, but what are some of the inherent challenges of working toward building a quantum computer using classical computers? To what extent do you need a quantum computer to build a quantum computer?

**KITAEV:** A quantum computer could simulate more complex quantum computers. I don't know, we're not there yet. We have to work with what's available right now.

**ZIERLER:** But that is to say that you can use classical computers. We do have at least the building blocks to get there, however rudimentary?

**KITAEV:** We can use classical computers. We can do some small-scale simulations of quantum systems on a classical computer, and that certainly helps. I'll give another interesting example of what could be done with a quantum computer. And John Preskill would give a better answer because he's done some real work on simulating field theories using a quantum computer. In high-energy physics, one can come up with models of strongly interacting field theories that are hard to solve theoretically. It would be very interesting to simulate them on a quantum computer. That could help make some theoretical prediction regarding processes at very high energies.

**ZIERLER:** Finally, looking to the future, for my last question, you mentioned before that you might caution students that 15 years from now, there might not be funding in the field. Given where we are today, where there's so much excitement, so many resources, and it's obviously a golden age for quantum information, what are some of the dangers or things to be mindful of to manage so that all of this excitement doesn't go away, even if these breakthroughs that are creating the excitement don't happen in the near term? How do we manage expectations and convey that patience is needed in order to see all of these things achieve fruition?

**KITAEV:** I don't know. I want to make a comparison with a different field, artificial intelligence. The idea of artificial neural networks is pretty old. It goes back probably to the 1970s, if not earlier. There was some initial hope to build an intelligent system using those networks, but it didn't work. In general, work on artificial intelligence in the 60s and 70s didn't give very strong results. Then, there was an artificial intelligence winter, as people call it. Some 15 years ago, a new spring came, and neural networks actually worked. People are using them a lot these days. But the technology was not ready during the early excitement period. I don't know exactly what the problem was in that particular case. But with quantum computing, I hope that the technologies we already have, like superconducting qubits, are mature enough to build a working quantum computer. In this case, the progress is likely to be continuous. But it might also be as if we were trying to make a mechanical classical computer in the 19th century. Then, the computer could have been built, I guess, given enough resources. But it wouldn't work very well. It wouldn't scale very well to solve real problems. In this case, we know that better technologies, like transistors or even vacuum tubes, just didn't exist in the 19th century. It might be the same situation with the quantum computer now. It might turn out that making a scalable computer these days is just not possible, and one has to wait maybe decades before better technology emerges. I don't know how things will turn out.

**ZIERLER:** It sounds like what you're saying is that a bit of historical perspective will serve the field very well in this regard.

**KITAEV:** Maybe. But it's hard to make predictions about the future.

**ZIERLER:** Alexei, on that note, it has been a great pleasure spending this time with you. I'd like to thank you so much for doing this.

**KITAEV:** It was my pleasure. Thank you.

[END]

#### Interview highlights:

- Error Correction Between Here and Quantum Computers
- The Source of Errors in Quantum Information
- Real and Toy Black Holes
- Quasicrystals at the Landau Institute
- From Station Q to Caltech
- Preskill and Topological Entanglement Theory
- New Topological Phases
- Testing Quantum Information
- Google and Qubits and Correlators
- Imagining the Benefit of Quantum Computers