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Guifre Vidal

Guifre Vidal

Senior Staff Research Scientist, Google

By David Zierler, Director of the Caltech Heritage Project
January 10, 2022

DAVID ZIERLER: This is David Zierler, Director of the Caltech Heritage Project. It is Monday, January 10, 2022. I am delighted to be here with Dr. Guifré Vidal Bonafont. Guifré, great to be with you. Thank you for joining me today.

GUIFRE VIDAL: Thanks, David. It's good to be here.

ZIERLER: To start, would you tell me your current title and institutional affiliation?

VIDAL: Last time I checked, I'm a Senior Staff Research Scientist at Google X, but I'm transitioning to being the same thing at Google Quantum AI. I'm switching teams at the moment. But anyway, it is all Google LLC, if I look at my employment contract!

ZIERLER: What is Google X?

VIDAL: OK, I called it "Google X" (its previous name) because otherwise people don't relate it to Google, but its current name is actually "X, the Moonshot Factory". It's the place I've worked for the last two years, that is, since I left academia. X is a semi-secret lab within Alphabet (the parent company of Google), where lots of interesting projects called moonshots are being explored.

ZIERLER: Are you specifically working in the field of quantum information at Google X?

VIDAL: My work at X has not been in quantum information. I wanted to work on something different for a while. I have explored how to repurpose Google's Tensor Processing Units (TPUs) as quantum chemistry supercomputers, to accelerate and scale up electronic structure computations. A TPU is this amazing piece of hardware that Google created exclusively to perform certain artificial intelligence workloads (training of, and inference with, large machine learning models). So a TPU is an application specific integrated circuit (ASIC) meant for just one particular computational task, machine learning. However, if you have a look at that hardware, you quickly realize TPUs could be used for things other than machine learning. In September 2019, I joined X to explore how to use these ASICs for computational quantum chemistry. Luckily, the project has been very successful, and for instance, a couple of weeks ago we managed to complete the largest density functional theory (DFT) quantum chemistry computation ever performed, using a quarter of a million orbitals. The results will be posted soon in the arXiv. More generally, we understood how to use TPUs to perform distributed dense linear algebra tasks (such as multiplying very large dense matrices, etc.) at unusually large scales, and how to apply them to simulating quantum computers and to perform computations relevant to quantum physics, quantum chemistry, drug discovery and nanotechnology. The broader implication is that we might have found a way to democratize supercomputing power in all these disciplines – not everyone can have access to, say, a supercomputer hosted in a National Lab or a private financial institution, but TPUs can be accessed online anytime by anyone.

ZIERLER: You mentioned that you left academia, although hearing you talk about this work, it sounds like fundamental research.

VIDAL: Yes, one could say I got away with murder, to some extent: I moved to industry hoping to tap in into their vast resources, which could lead to broader impact, but I continued to do something that indeed could be mostly characterized as fundamental research. However, compared with my work in academia, which was in quantum information, condensed matter and theoretical physics, at Google X I was "forced" to keep an eye on potential industry applications. As a result, I have found myself on discovery calls with big pharma, chemical and even financial companies that are potential users of the TPU-based computational solutions my team has built.

ZIERLER: In the race, if that's the right word, to develop a quantum computer, how do you understand Google's approach that might be separate from IBM, Honeywell, or Amazon?

VIDAL: That's a tricky one. I don't know how much I'm allowed to talk about Google's approach beyond what's available out there. I need to really figure it out. I've started interacting with my new team, Google Quantum AI, the team building Google's quantum computer. I'm going to be working with them starting at the end of February. But I'm not sure how much of that I can discuss here.

ZIERLER: What were the circumstances of you leaving academia? What were your motivations in joining Google?

VIDAL: As an academic researcher, I was in one of the best possible Theoretical Physics institutes one could possibly wish to be in. The Perimeter Institute in Canada is a truly remarkable place for theoretical physics. When I left Perimeter Institute to move to Silicon Valley, I left behind lots of good friends and genuine scientific excitement. [Sigh] We were indeed doing exciting work. However, there was something I could not ignore: in recent years, we have seen how quantum science and technology has stopped being the exclusive property of academic research institutions. Actually, most quantum computers are currently being built by big technological companies. But let me first go back in time. My relationship to quantum computers is ancient, since the beginning of my PhD in Barcelona in 1996, that is, literally since last century. Back then, there were no quantum computers, only big ideas, and the promise that it would only take 10 years to build the first one, guaranteed. After completing my PhD, I went for a first postdoc with Ignacio Cirac in Innsbruck, then for a second postdoc with John Preskill at Caltech's Institute for Quantum Information, where I remained until 2005. And guess what, in 2005 they were still saying quantum computers were only 10 years away, guaranteed. But it was clear that the technology wasn't ready. So I decided to branch out into other topics instead, such as tensor network algorithms for condensed matter and high energy physics problems, while waiting for quantum technology to catch up. Let us forward back to the present, or rather to 2019, that was twenty years after the completion of my PhD, and quantum computers were finally here. Not yet a full-fledged quantum computer with quantum error correction and all, but for these noisy quantum devices with programmable gates that in the short term can already be used, for instance, to emulate other quantum systems. That was exciting! I thought it was perhaps finally time for me to go back into quantum computing. However, what had traditionally been a little bubble in academia, was then being taken over by Big Tech, with immensely more resources than the academic world will ever have. So I figured that going back to quantum computing meant transitioning to industry. Then I accidentally ended up at Google X repurposing TPUs for computational chemistry – I am not sure how I can explain that, but in a few weeks, I will be with Google Quantum AI [laugh]

ZIERLER: Is that to say that, in some ways, the engineering is now finally catching up to the theory?

VIDAL: Yes, indeed, but only to some extent. The theory involving thousands of error corrected qubits is so far ahead that I do not expect engineering will catch up in my lifetime. However, there are already these cute shiny machines with all sorts of knobs that you can call quantum computers, if you start stretching the definitions. They are not yet fault-tolerant, and they are very noisy. That's fine. We have machines right now with 50-plus qubits, that you cannot simulate with a classical computer in any simple way that we know of. And this is only going to get better. We don't have a quantum computer yet capable of any "useful" computational task, such as factoring numbers large enough so as to disrupt current cryptographic schemes. Of course, they'll say we will have one in 10 years, guaranteed. But there are finally these quantum devices today, quantum toys we can start playing with, that will only get better over time. It's no longer pure speculation. As a theoretician, whatever you may propose we could do with this toy today may help inform what the toy should look like in the next iteration. I feel more comfortable, and enjoy it better, when I put energy into discussions that connect with reality and may even have a real impact on quantum technology.

ZIERLER: It sounds like, for you, there isn't a clear line that delineates an era before quantum computing and an era after, that by some definitions, we're already there. The question is, how much better is it going to get?

VIDAL: It doesn't bother me, as a scientist, that full-fledged quantum computers are not yet there. There's something that exists today that is based on quantum, so there's a machine, a complex system that I can start poking. If I was a businessperson…—wait, am I a business person now? I'm not sure, I will have to ask my manager. If I was trying to extract value out of a quantum computer for business purposes, I would say, "Oh, this is 100% hype. There's nothing useful yet. Quantum computers are not yet surpassing classical computers in anything useful for any sensible definition of what useful means." And that's not going to change any time soon, I don't think so. There are of course all these articles in popular news, or even better, in specialized online technological magazines, listing dozens of breakthrough applications of quantum computing in the business world. [Laugh] We do not yet have any practical commercial application of a quantum computer, that is, besides of course the recent realization of a time crystal that can act as a time machine.

ZIERLER: This is a discussion about commercial applications. What about applying quantum computation to physics itself, some of the unending challenges, theoretical and experimental, in physics?

VIDAL: On that side, I think we've already seen spectacular progress. I'm joining the Google Quantum AI team, so I'm going to say something nice about the Google Quantum AI team, but I think I would say the same even if I was not joining them. I've seen a series of very impressive papers demonstrating the use of the quantum computer they have for simulating quantum physics. I'm sure other companies have similar results, but the ones I know better are coming from Google. What I'm seeing there is, in the context of simulating quantum systems, this quantum computer has many more knobs than other previous quantum emulation platforms, say experiments with cold atoms in optical lattices. At the very least, the quantum computers we have today are more versatile when it comes to simulating quantum systems than we've ever had before, so that alone is very interesting.

ZIERLER: What might we read into the artificial intelligence part of Google's Quantum AI initiative? How do you see artificial intelligence as part of the quantum computation revolution?

VIDAL: I think that's a very smart question to ask, but unfortunately, the answer cannot match the level of the question. [Laugh] I think that might just be a bureaucratic mistake. I don't think there's much about AI in Quantum AI. But I am not sure.

ZIERLER: Now let's go back. What years were you at Caltech for your post-doc?

VIDAL: I should know this. 2001 to 2005. Yes, I think that's correct.

ZIERLER: And before you had been in Barcelona and Innsbruck. When was that?

VIDAL: I did my PhD at the University of Barcelona 1996-1999 and a first post-doc at the University of Innsbruck 1999-2001.

ZIERLER: Was all of your research in quantum information?

VIDAL: Yes, from the very beginning. I worked in quantum information during my PhD in Barcelona. Then also during my first post-doc in Innsbruck. And then also during my second post-doc at IQI with John. I was at IQI for four years. During that time, I started to be more and more interested in other topics, especially in understanding the structure of entanglement in quantum many-body physics; also, on how to exploit that entanglement structure to come up with new classical simulation tools for quantum systems. As I said, I felt that quantum computers were not going to exist for a long time. So I thought: "Let's do something else in the meantime". This ended up leading to improved numerical approaches for simulating condensed matter systems. If you were talking to Frank Verstraete earlier today, he may have mentioned tensor networks. Tensor networks, as far as I'm concerned, are what happens when you try to understand how to apply to condensed matter problems the entanglement lessons learned in quantum computing.

ZIERLER: In your PhD, did you think that quantum computing was going to happen faster than it actually did? Was this a realization at Caltech that it would be a longer time coming?

VIDAL: I've never been very good at predicting the future. I didn't mean to decide for the field what was going to happen, but I was concerned about what I was going to be doing. Throughout the PhD in Barcelona, first post-doc in Austria, and second post-doc at Caltech, I worried that theoretical quantum computing was too removed from practical reality. Another concern I had was that perhaps quantum information was too much of an artificial bubble, full of excited young researchers (myself among them!) agitatedly asking and answering lots of new questions nobody else had previously asked, perhaps because, after all, they were not really that important (how many inequivalent classes of entanglement can we have in a system with 10 qubits? What is the channel capacity of this n-qubit quantum operation in the large n limit? What happens if we compute the partial transposition of this extremely fine-tuned, bipartite mixed state?). I remember thinking that, if what we were learning about quantum entanglement was truly useful, one might be able to find applications outside quantum information. Yes, my colleagues could not stop coming up with new stories involving Alice and Bob, and I hated that. How about trying to answer questions that had been asked before by other people in other fields (instead of working on the next episode of an increasingly convoluted Alice and Bob series)? You can always fall into this situation where you generate new questions because you know the answers, and you create your own bubble. But I think more interesting than that is when you can impact or contribute to answering questions that other people have been asking in another field.

ZIERLER: What are some of those questions in the other fields?

VIDAL: For instance, consider a two-dimensional strongly correlated quantum material that you would like to understand from first principles or to build some cool technology out of it (say in the context of searching for a room temperature superconductor). A first step could be to use a classical computer to try and simulate it. But then you have to fight this exponential growth of the Hilbert space, of the vector space of quantum wavefunctions. The number of computational resources you need to study quantum many-body physics grows exponentially with the number of quantum spins. An important question was back then, can we do better than exponential? Of course, for some particular problems there were already useful methods, such as quantum Monte Carlo, which is very useful except in a wide range of interacting systems where it is afflicted by the sign problem. Tensor networks, which are based on understanding quantum entanglement, really ended up making a difference. Actually, to be fair, Steve White's DMRG is already based on a tensor network, a matrix product state, and precedes Frank Verstraete's work and my work on tensor networks at the IQI by more than 10 years; but it was not until Frank, myself and others understood the entanglement structure of ground states that we could start making sense of tensor networks and generalize DMRG from ground states to time evolution simulation and also to other structures such as PEPS or MERA.

ZIERLER: When did you first become aware of John Preskill's work? Would it have been in graduate school?

VIDAL: Oh, that would have been natural, given how impactful John Preskill's work had already been. However, during my PhD in Barcelona, I was not very aware of what was going on internationally. I could not even conceive that the people who were signing amazing papers were actually real people you could meet with and talk to. I had very limited visibility into the world out there (literally: my office did not even have a window). I remember being in shock when I met Ignacio Cirac, and later even Michael Nielsen, Charlie Bennet or Sandu Popescu, at some European workshop, near the end of my PhD. It turns out that they all were real people. Amazing! As for John Preskill, he was not at that workshop, so as far as I was concerned there was no evidence that he existed, yet. I think I might have finally understood who John was when it was time to look for a second postdoc and someone recommended that I should write to him and ask if I could join his group at Caltech.

ZIERLER: What were you doing at the time you were inspired to write to John? How did you make the intellectual connection that Caltech would be a good place for you?

VIDAL: I wouldn't call it an intellectual connection. [Laugh] During my PhD I was very lucky to meet Ignacio Cirac, who offered me a postdoc in his group in Innsbruck. Once in Innsbruck, I started to meet some more international researchers. But that was in Austria, and there was no John Preskill to be seen there yet. However, my first postdoc was nearing its end and I panicked, "What do I do next? I've been trying to avoid this question forever, but I have to start applying." That's when I applied to Caltech. It was in order to attend job interviews in these places that I first visited the US. And finally met John!

ZIERLER: Going all the way back to undergraduate, to get a sense of the maturity of quantum information, was that something that was even available for you to study as an undergraduate?

VIDAL: Nope.

ZIERLER: How do you go from an interest in physics to quantum information for graduate school?

VIDAL: This I owe to my PhD advisor, Rolf Tarrach, who incidentally decided to quit physics shortly after being my PhD advisor. [Laugh] Back then Rolf was a high energy physicist at the University of Barcelona. He was the one in the physics department who said, "Now I'm going to do some quantum computing." He told me, "You asked me if I had a PhD project for you. We're going to do quantum computing." I was like, "Oh, OK. I don't know what that is, but fine." Later, I realized he didn't know what that was, either. But that's how I got introduced to quantum computing and quantum information, through my PhD advisor who was curious about the topic.

ZIERLER: What were your initial impressions when you arrived at Caltech?

VIDAL: It's good you asked this. My initial impression was, "Wow, everybody's so young here." All the students looked really young. Like prodigy kids. But once in the physics department, I couldn't find John Preskill. Nobody even knew him! After some questioning, some teachers helped me understand that that was not Caltech, that was Pasadena City College. [Laugh] I crossed the street and finally went to the other Caltech, the real one.

ZIERLER: You found John then.

VIDAL: I must have, eventually. I don't remember. I only remember Pasadena City College. Which I thought was really cool, but something was clearly wrong. Anyway, yes, Caltech. John. John had the vision to create a place, then called the Institute for Quantum Information IQI, and to hire interesting young people for it, then apply a hand-off approach to it and the philosophy of, "Let's see what happens." Recently, John and I finished a paper, and that's the first paper we have together. I was in his group for four years, and we did not write a single paper together, back then. John was doing his thing, collaborating with some people in the group, mostly his PhD students, and giving the rest of us plenty of space.

He was like, "You guys do whatever." We'd have our group meetings. Once a week, he would check that everybody was still up and kicking, nobody had disappeared or anything. He asked what we wanted to report each week. But he's not the type of person who needs to control everything and everyone, and to have his name on all the papers produced in his group. Instead, he created this space where we would have freedom, surrounded by very talented people, all of which were of a similar age. It was only natural for us to start interacting and collaborating. I credit John with the ability to create that space for a number of young people that he was selecting somewhat carefully, I'm guessing, well, except for me, of course. [Laugh]

ZIERLER: Was your sense that IQI was essentially a bunch of post-docs and John? Were there other faculty who were around a lot when you were there?

VIDAL: We had other faculty around. I clearly remember Alexei Kitaev, he was there already, he is hard to forget! And Leonard Schulman was also there sometimes. Probably there were more faculty, and plenty of faculty visitors, but I forget who. We also had John's PhD students, of which there were perhaps as many as post-docs.

ZIERLER: What were the big ideas? Frank used the metaphor over and over again that this was the golden age, and there was low-hanging fruit in quantum information. I wonder if that accords with your memory, and if so, what were the low-hanging fruit at that point?

VIDAL: I guess it makes sense to put it that way. I think Frank is being uncharacteristically modest. [Laugh] He's probably been studying that answer for a while. [Laugh] I miss Frank. It is true that in hindsight, what we did back then might have been technically simpler than subsequent refinements, but a lot of original ideas were generated back then. And so much has been done later on building on what we did. When I say we, I mean people there. Frank, Dave Bacon, Patrick Hayden were there…

ZIERLER: Andrew Childs is a name that came up.

VIDAL: Andrew Childs was there a bit later, but yes. And Yaoyun Shi, Wim van Dam, Debbie Leung. Also, Sergey Bravyi. Sergey's one of my favorites. I am sure I am still forgetting people. Luming Duan! A lot of work has been done since then on top of what we did. But I think it is fair to say that we did a lot of good work in terms of demonstrations of principle, working on problems that were conceptually new or had significant novelty to them. (Fine, Frank wins again: low-hanging fruit). For instance, the work on tensor networks. Sure, you can later elaborate on it, you can have 25 PhD students working on refined details or ever bigger constructions, but the essential breakthrough that opened up the field happened while we were at IQI.

ZIERLER: Where among the post-docs was there collaboration, and where was there competition among you?

VIDAL: I only remember collaboration. I remember being a bit overwhelmed by how much potential for collaboration there was, how many things I could do. Back then, I had mostly been working as a single author. I was trying to force myself to not do just that because I could see how much my colleagues knew that I could learn. I never felt there was competition there. OK, maybe Frank and I were competing, but I also learned a lot from him. The problem is, we were sharing an office, which was not very large – that is, compared to our egos. Or should I say compared to his ego? [Laugh]

ZIERLER: Was your sense that IQI was doing things that weren't being done at peer institutions that had similar quantum information programs?

VIDAL: I actually was not aware of that, not back then. A lot of what I'm saying now is in hindsight. I realized how special IQI was later on, after I have been part of or visited many other groups. Back then, to be honest, the most special part of IQI for me was the free pizza on Wednesday night – or whenever it was. Wow, we had free pizza, and also Thai food, Mexican food, I was unable to focus. I thought the rest of IQI was just normal.

ZIERLER: You said you were almost hyperactive in your productivity as a post-doc at Caltech. What were the things you were working on? Why was there so much to do?

VIDAL: There were so many interesting problems to look into. On the one hand we had quantum information problems (yes, often involving Alice and Bob) regarding entanglement measures and entanglement capabilities, channel capacities, separability studies, etc. Then there were applications of entanglement measures to condensed matter systems, starting with quantum spin chains. One thing I did with Alexei Kitaev and two other collaborators was understanding entanglement entropy in a quantum phase transition, which Cardy and Calabrese also worked on subsequently. Then there was all the work on using tensor networks to simulate quantum computations and establishing the relationship between entanglement and quantum computational speed-up. Something else was to relate that to ground states of local Hamiltonians and then understand how to efficiently simulate condensed matter systems. I would have to go back to the arXiv for more examples. I really forget these things. But I remember having to choose every day what to do next because there were several problems I'd like to work on, and I only had that much time.

ZIERLER: Maybe the question, then, is, what are the boundaries? When you're a post-doc at IQI, what are the boundaries of the kinds of things that would be exciting or relevant to work on in quantum information?

VIDAL: I don't know the answer to that. Which, I guess, means that to me there were no obvious boundaries at IQI. I do remember going to John once and asking him, "What should I be working on? What's exciting?" He said, "What you're doing." [Laugh] Which you can take it as a compliment "OK, if that's the advice from the expert, then I must be doing fine!" But John can also be a very sarcastic person, so you could interpret it as, "Go away, Guifré, I'm not in the mood." [Laugh] "And, by the way, I'm just going to embarrass you by saying that what you are doing is interesting." There was a lot of freedom to explore your own way and, somehow, I think we managed to use that freedom properly.

ZIERLER: Was the overall research approach at IQI really just basic science? Were people thinking about applications that early on? Is there a general sense that, "We're working to build a quantum computer"? Or does that only come later?

VIDAL: I don't think we were overly concerned with building an actual quantum computer. Alexei Kitaev was making his pioneering contributions to topological order, the type of thing that has been super influential later on (Google is currently trying to implement Alexei's surface code in their quantum computer!), but back then, that was too removed from existing technology. Daniel Gottesman was also worried about error correction and encoding qubits in real systems such as oscillators. I wouldn't say, however, that there was major concern for how to realize things in practice.

ZIERLER: Tell me about these famous Wednesday group meetings. What was the atmosphere, what was it like to share ideas, how did you get new ideas on things to work on as a result of hearing from your colleagues?

VIDAL: I remember that as a bit of a pain. We would talk and talk. It was stressful because at the same time, you could see the free food arriving. [Laugh] They would bring the pizzas, bring Thai food, Indian food, Chinese food, and then I was completely unable to concentrate on anything else. It was like, "The food is here. Why are we still talking?" Other than that, it was nice but super nerdy.

ZIERLER: What do you see as your big contributions while you were at Caltech?

VIDAL: Looking at work that has follows from that, I would say that coming up with the time-evolution simulation techniques for MPS, understanding the structure of entanglement in quantum phase transitions and proposing the MERA tensor network are probably the three big contributions while I was at Caltech. Of those, I like MERA best, as I think of it as a modern reformulation of the renormalization group. The renormalization group, due to Wilson, Kadanoff, etc, was of course already established in physics by decades of good work in statistical mechanics and field theory, but the quantum version of it, I would claim that it was not well-understood yet, at least in real space. Having been able to contribute that in the form of a tensor network is something I'm proud of.

ZIERLER: When you moved to Australia, was the idea that you would build your own program and model it, to some degree, on IQI?

VIDAL: Oh, no, not really. I'm not like John. John has a lot of patience and he's calmer. I'm more like, "OK, let's do something. Fast. Now." I am also a bit of a micromanager, I need to know exactly what is happening in my group. But yes, I did build a research group. I first hired a few PhD students (in Australia, good PhD students dif not cost you money, they would come with their own grants). Then I was overwhelmed by how much work it took to supervise them. So I had this very clever idea, best idea ever, which of course did not work: I applied for several grants, then used the money to hire a layer of post-docs that would shield me from my students. Obviously, that totally backfired: now I had to supervise both the students and the post-docs. That really meant the end of my "independent contributor" phase. Anyway, no, I was not trying to emulate the IQI. Or, if I was trying, I certainly failed big time!

ZIERLER: What were the circumstances of you moving to the Perimeter Institute?

VIDAL: I think weather-wise, a complete miscalculation.

ZIERLER: [Laugh] Did you not know where Canada was on a map?

VIDAL: No, I knew, I knew. And I've never regretted moving to Perimeter. Australia was great, both as a place to live and to work as a researcher. I loved it so much there that I even became Australian! However, Perimeter represented a better opportunity in terms of being connected with the bulk of researchers in North America (these were pre-pandemic times, when you really had to meet people in person in order to have a research discussion). Perimeter was, and is, some sort of Mount Olympus for theoretical physics.

ZIERLER: Last questions for our talk. Looking to the future, I know that you can't talk so much about what's happening at Google, but I wonder, more broadly than that, when there are going to be these major advances in quantum computers, where will you see Caltech or IQI DNA in this endeavor?

VIDAL: Here's a detail. In the Google Quantum AI team, we have Sergio Boixo, Dave Bacon, Alexei Kitaev (part time) and myself, all of whom were at IQI. In the quantum team at AWS you find Fernando Brandao, John Preskill, Ashley Milsted, etc, also from IQI (or IQIM). Yaoyun Shi is with Alibaba quantum team, and so on. A large number of people who had been in the IQI(M) at some point in their careers are now part of the quantum computing team in some major tech company. In genetic terminology, I am not sure how to characterize the IQI DNA. Can we say that former IQI members are like a virus, spreading over and corrupting the quantum computing community? I wonder what Frank Verstraete said to this one…

ZIERLER: Finally, what do you want to accomplish next? What's most exciting for you?

VIDAL: I look forward to seeing what's next in terms of using quantum computers as quantum emulators, that is, to simulate other quantum systems. Getting to practical quantum error correction is a natural, major next goal for quantum computers, one to which I hope to also be able to contribute as part of the Google Quantum AI team. However, my money is on us first being able to use the current noisy quantum computers, perhaps even without error correction, to make progress in understanding quantum phases of matter. Can we perhaps use a noisy quantum computer to finally understand how to design room temperature superconductors? Or make better batteries or solar cells? I am also excited about recent progress in classically simulating quantum devices and computational methods for quantum physics and chemistry, e.g. using tensor networks. As it turns out, in order to understand what quantum computers can do for us better than classical computers, we need to also make progress with classical algorithms, and this results in a stimulating and productive conversation between quantum and classical techniques.

ZIERLER: Well, Guifré, thank you so much for spending this time with me. I really appreciate it.