 Thank you IBM for saying up this conference, which I've enjoyed a lot. So today I'm going to tell you about some work that we've been doing at Google, which has also been in collaboration with our previous speaker, Garnet, and our next speaker for Microsoft, Nathan Wiebe. And that work concerns quantum simulation with low depth circuits. I'm going to give this talk at a fairly high level, so hopefully most of the room will sort of follow along. But if you're more interested, I would refer you to these three papers, especially the last two, which go into a lot more detail. So I'm going to start with a rather, I think, uncontroversial statement that if we're going to do something interesting in the near term, it's going to need to be with a rather low depth circuit, just due to limitations of fidelity and coherence and whatnot. And as we heard in some prior talks, there's been some work in sort of identifying what it might take to get to the point of classical attractability, a particular proposal that our group is interested in. Concerns simulating random circuits, as McBremner talked about. We think that you can get to this classically intractable regime with maybe 50 qubits, gate depth 40, and about 99% fidelity. However, the supremacy experiment is not intrinsically useful. And of course, we'd really like to be able to do useful things with quantum computers. So the supremacy experiment, I think it sort of gives us optimism that with a low depth circuit we can prepare rather complex quantum states. So the question then becomes what's sort of the value in being able to do this. And sort of rhetorical answer to this is that such states can provide descriptions of complex quantum systems, which maybe you couldn't model classically. So of course, systems of interacting electrons of the sort that Garner and Bella talked about before are such systems that are often very hard to model classically. And often, in many instances anyways, describing these systems is a bottleneck for computational chemistry and material science. So to give some sense of this, I mean, exactly using known methods, exactly computing something like methane might take seconds on a laptop, maybe minutes to do ethane. It's gonna be quite difficult to do propane to chemical precision. It could easily take days. Now whether this is going to be asymptotically exponential scaling or not is sort of something that Garner and I talked a lot about. It may not be, but the fact of the matter is that going to even slightly larger systems than this today to high precision is essentially intractable. And to send, especially simulating something like this, a superconductor simulating this at the atomic level would be extremely challenging. So the question sort of becomes, well if we wanna model these things and we can create complex states, how do we make the right states? And the idea that I think is very popular with people in this room is that that of variational algorithms that you can essentially train quantum circuits to prepare these states in a way sort of similar to a neural network. So I'll be talking a bit about that today. And specifically, I'll be focusing on ways that we can use insights from chemistry and Hamiltonian simulation to sort of do this with lower depth circuits. So I should go through this very quickly. We're interested in solving the Schrodinger equation for molecules. There's Coulomb operators and particles have connect energy. We're usually working in the Born-Oppenheimer approximation, which means that we're looking at the electronic wave function and the potential of the nuclei. So there's an energy surface that is sort of parameterized by where the nuclei are. In the case of molecular hydrogen, it's just the bond length. You've seen a lot of plots like this already at this conference. And energy surfaces are very important to chemists because they teach us about chemical reactions. In particular, changes in energy on these surfaces are responsible for determining the rates of chemical reactions. And we're especially obsessed with what we call chemical accuracy, because that's the accuracy that you need in the energy surfaces to compute the rates accurate to within an order of magnitude at room temperature. And it's especially achieving this accuracy, which is, in many cases, classically intractable. And that's especially so for the sorts of systems Garnet talked about, certain strongly correlated systems, especially those involving transition metals. And so what's at stake here is essentially a paradigm shift from essentially a qualitative and descriptive way of using computation that's sort of done after the experiment to a protective and quantitative paradigm, which would really allow for in silico material discovery, that sort of thing. So I debated whether or not I needed this slide at all. I'll go through it rather quickly. I'm sort of interested in variational algorithms. These are based on the variational principle. They are especially straightforward to understand in terms of preparing ground states of systems. The idea is that you have some parameterized quantum state. You create with a circuit. You measure its energy, and then you use some classical outer loop to try to find parameters that minimize the energy. So the first step in this is you need to come up with a parameterization of a circuit that's going to prepare the state. And I write cleverly here because I'm of the belief that it's very important to sort of include insights from say chemistry in order to correctly structure these circuits. Now popular ways of coming up with these circuits include, for instance, taking a low depth trotter approximation to the 80 back state preparation. Another idea that's been talked a lot about is unitary couple cluster, which is basically a way of using perturbation theory in order to come up with sort of a good initial guess for what terms you should use in this thing and what initial parameters you should take. But anyways, once you've come up with that, you need to prepare some reasonable classical guess state. Maybe it's the classical mean field state. It should reflect a good classical approximation to the problem. So you prepare that state presumably easily. You then prepare this parameterized circuit. You get out a parameterized state. You measure its energy pretty straightforward. And then you hopefully are close enough to the ground state that you can use some greedy optimization like gradient descent to suggest new parameters. And the last couple steps here are then iterative. So quickly, I was going to tell you about this rather old experiment that we did. It's almost two years old now. So this was a simulation of molecular hydrogen on superconducting qubits. We did two different algorithms. One was the phase estimation algorithm I'm not talking about. But the red dots were the result of our variational experiment, which as you can see performed fairly well for this system. But I think perhaps the most interesting part about this experiment was actually if you take a close look at the errors in the results that we got. So in red here is the errors in the red points on the left as a function of the bond length. And so in this case, we essentially determine the optimal parameters experimentally by using this classical outer loop. But because this is a small system, we also knew what the theoretically optimal variational parameters were. So we also ran the circuit at these theoretically optimal parameters. And when we did that, we get the green dots. And you can see that the circuit run at the theoretically optimal parameters gave energies that were almost in order of magnitude worse than the circuit ran at the parameters that were determined to be the optimal experimentally. So that's pretty easy to understand what's going on here. Variational algorithms are robust to certain types of control errors. For instance, if one of your parameters is a rotation angle and you're systematically over-rotating something, then in the course of the classical optimization, instead of finding theta star the optimal parameter, you'll find theta star minus epsilon or something like this. The optimization will drive you to the correct angles. So this is one of the things that this robustness to systematic errors, it's one of the things that I think sort of inspires hope that variational algorithms might have a chance. Of course, these are heuristic algorithms, which differentiate them from a lot of the simulation approaches that we talked about earlier today. OK, so now I'm going to sort of dive into ways that actually, no, I've got a slide ahead of myself there. OK, so in the experiment I just showed, we actually did use unitary coupled cluster. And unitary coupled cluster, like other structured onsots like Traurized, AdiVac, state preparation, it does require a large number of arbitrarily connected gates in order to implement. These methods usually have some cost that's proportional, polynomially, in the number of terms in the Hamiltonian. And in the normal representation, the Hamiltonian has end of the fourth terms, which I'll talk about. And so that leads to quite a high cost. Now, I probably can't give this talk here without at least mentioning that IBM had a recent result. They did a variational chemistry experiment as well, where they sort of surpassed this problem with the very expensive variational onsots by proposing to use what they called a hardware efficient onsots, which essentially just used the gates available on their device. In fact, I took this picture from their paper, it was sort of layers of entangling gates and then parameterized rotation gates. And this allowed them to extend their approach to larger molecules and what we had looked at before. We went up to three qubits previously, they went up to six qubits and they've now looked at bigger things than molecular hydrogen. Which is, I think, very exciting and allowed them to sort of look at a lot of the more challenging parts of this experiment, like what happens when you have more variational parameters, they study new optimization techniques and new representations. But something which I think we also need to keep in mind here is that accuracy is very important in these quantum chemistry simulations. And in particular, these experiments do not surpass the accuracy of classical mean field methods, which means that I believe anyways that this is sort of evidence that we need to think carefully about how to structure the variational onsots because in this case there is a product state that has lower energy than these results. And there's other challenges too, which just come up sort of generically and make the experiment much more difficult. So one thing, for instance, is that a very large number of measurements are required. If there's end of the fourth terms, then naively it looks like you're gonna need end of the eight measurements. Now in practice it's probably a bit better than that, but this is still a lot of measurements. Now the measurements is a classical resource. It corresponds to how many repetitions of the circuit that you need to make. But still, this is a real cost. I mean if you're looking for robustness to systematic errors and you have to repeat your experiment 10 million times, there's going to be drifting parameters, which sort of negate the effects of that. So that can be a real challenge. So what I'm gonna talk about in the rest of this talk is basically ideas that sort of build on this idea, IBM sort of started of this structured variational onsots, which, or sorry, hardware efficient variational onsots, thinking about an onsots that fits on the hardware that you have and is sort of very efficient in that sense, but uses insights from chemistry to come up with a good initial parameterization. All right. Okay, so to explain this, I need to go back to the basics somewhat here. So whenever you're simulating a wave function on any computer, you need to discretize it somehow. So if you have eta electrons, you're gonna need to somehow confine them to a grid of endpoints, which means that your wave function can potentially span n choose eta possible configurations. Now if you're going to simulate these things classically, you should be very worried about that because the number of configurations is growing combinatorially. So for this reason, classically it's very important to choose a basis where the wave function is, where the ground state or whatever state you're interested in is compact, meaning it has support on the fewest number of configurations possible. And because of this, chemists classically have long used molecular orbitals, which are a very acceptable solution to this problem. They're very compact. They are obtained from the mean field solution and they sort of look like this. I mean, this is, for some molecule what some molecular orbitals look like, they're usually made from atomic orbitals. And I should mention that normally we're gonna simulate these things in second quantization, where each qubit basically encodes an orbital. It's one if there's an electron there and zero otherwise. And so once you've made discretization choices like this, in second quantization as Garnet pointed out, you get a Hamiltonian that tends to look like this, where these things are fermionic operators, Adag or P-A-Q excites from orbital Q to orbital P. This would be the one body term containing information about the kinetic energy and external potential. This is the two body term that just comes from the Coulomb interaction between electrons. Now, there's different ways you can come up with these values of these coefficients here, but by far the most common way in chemistry is what's called the Galerkin discretization, which uses integrals over the basis functions that look like this. Now, if you're concerned about, as I am, the end of the fourth terms in this Hamiltonian, you might come up with some ideas just by looking at the form of these integrals. You might notice, for instance, that if you were to choose basis functions which were spatially disjoint, meaning that they didn't overlap in space, then instead of end of the fourth of these, there's actually n squared of them. So these phi's here are the basis functions, and you might notice that phi P and phi S, they have the same electron position variable. So phi P doesn't equal phi S, or phi Q doesn't equal phi R, and the basis functions are spatially disjoint, then the integrand is zero everywhere. And this would lead you to a Hamiltonian that looks like this, and you would have a diagonal potential. However, for the case of spatially disjoint basis functions, you run into problems in the one-body term, because this would also diagonalize the one-body term, and that can't be correct. In fact, it turns out this sort of discretization is kind of pathological for Galerkin discretizations. Of course, there are other discretizations you can choose. For instance, a finite difference discretization would be fine. However, there are reasons that chemists tend to prefer the Galerkin discretizations. For one thing, they have variationally bounded basis set air. And if you're using a variational method when you're trying to push the energy down, it's nice to know that your basis set air is variational from above. Whereas finite difference discretizations, for instance, don't have this property, and there's also other convergence issues related to them that you might or might not be concerned about. So I'm now gonna tell you about a way that you can use basis functions, which do admit this nice discretization, which have some other nice properties, which will also lead us in some way to gain n squared terms. And this is to use the plane wave basis, which I think should be kind of familiar to many people here. Plane waves are what you think. In my equations here, this omega is a volume parameter. It's just the volume of the unit cell over which the plane waves are defined. And so the wavelength of the plane wave is sort of this omega to the 1 third if it's the fundamental harmonic. So using the plane wave basis, it turns out that the Coulomb operator isn't end of the fourth. It has end of the third terms. You can write it like this. This is an exact representation that you would get where you two have done the integrals that I showed on the previous page. And interestingly, it has this nice analytic form. There's, the integral can be evaluated exactly. I won't get into it in this talk, but if you follow the sort of research that say Andrew and Robin spoke about this morning, you're interested in these signal processing algorithms. There's huge advantages to the fact that these coefficients are analytical because it means they can be computed on the fly. And in that paradigm of the simulation, that's very beneficial. But the reason we have end of the third terms just comes from conservation of linear momentum because plane waves are eigenstates of the momentum operator. And the two-body term, the electron-electron interaction is translationally invariant, so it needs to conserve linear momentum. So you can choose that you're getting plane wave P and Q excited, and you can choose maybe to take it from one place, but this constrains the other parameter because they all have to add up. The other thing to keep in mind is if you're using plane waves, you now have a periodic representation of the system. You can think of the periodized Coulomb operator, which looks like this. This is just the Fourier series of the one over r potential. And so, you know, that is of course ideal in many cases for periodic systems, like crystals. If you wanna do a sort of atomistic model of periodic systems, instead of, say, mapping it to a Hubbard model, which was the sort of thing that Bella was talking about earlier, that can be very good for gaining, say, qualitative understanding of materials. But if you wanna do, say, material design where the properties you're looking for depend sensitively on the particular structure, you're going to need to do some sort of explicit representation of the periodic system, the crystal, or whatever it is that you're interested in. So I should also mention, though, that if you don't want periodicity, you have to do a little bit more work to remove the images that are created by plane waves. So for instance, you need twice as many plane waves for each non-periodic dimension, so you'll need twice as many more if you're looking at graphing in steve diamond, or four times as many more for a polymer, eight times as many more for a single molecule. Now, this is a different issue than the intrinsic basis at discretization there. I was just talking about reducing periodicity. So you might be kind of skeptical that, you know, some basis functions like this are as good at representing the system as some very fancy thing that looks like this, and that's fair, but there's a lot known about this. So in particular, the basis at discretization error converges is determined by cusps in the wave function. This is sort of intuitive, but it was proven very rigorously 60 years ago. So we know that is the case. And so in the first moment of the wave function, if you're thinking about the density, and mind you, in most classical methods, like DFT or mean field methods, the only cusps you have appear in the density because those methods aren't correlated, so there's no electron-electron cusps. But certainly, there's cusps at the nuclei. Now, the Gaussians, they are centered on the nuclei, and as you add more Gaussians, they can get sharper and sharper, and they can resolve those cusps. And in this way, they can suppress the basis at error in a way that's very chast factory. It's, you know, it's taking care of. Plain waves do struggle a bit with nuclear cusps. However, in practice, you're almost always going to use pseudo-potentials when you're modeling these systems. For instance, in pretty much all the work in this area, including, say, Microsoft's work counting T gates for ferred oxen, or even in IBM's recent simulation experiment, an active space is used. And in an active space, what you do is you recognize that, say, only certain electrons are important for bonding. For instance, the valence electrons, and usually the core electrons, they can sort of be treated classically. So you take those core electrons, you freeze them, and you treat them with, say, a mean field method, which then dresses the nuclei with some external potential. This softens the Coulomb potential and restores analyticity to it. And for that reason, the wave function no longer has a cusp there, and the Fourier transform will suppress those errors extremely effectively. So for neither plane waves or goushians is the nuclear cusp really the thing to worry about, despite what is often people's intuition. By far, the bigger issue is the electron-electron cusp. So this occurs at all points in space, where two electrons might overlap. And goushians are centered on nuclei. The electron-electron cusp is everywhere. So goushians don't have any special advantage for resolving this electron-electron cusp. And yet, when you're interested in a correlated calculation, in fact, in many cases, most of the correlation energy comes from resolution of this electron-electron cusp. So it's very important. And it turns out that all single particle basis functions are sort of similarly inadequate at resolving this. And they lead to algebraic convergence of the basis set error, which goes like 1 over n. So for this reason, asymptotically, there's no advantage in whether you're using plane waves or goushians for single molecules or whatever. However, at small sizes, it certainly is the case that there are a number of systems where you're certainly going to need more plane waves than goushians. The converse is also true. There are also some, you know, there are periodic systems, especially, where plane waves are near the ideal basis to model the system. It sort of depends. But I could give sort of a whole talk about this. It's a long conversation. But another thing to keep in mind is that often people aren't just doing single-point calculations. They do a series of calculations, and then they extrapolate how the basis set error is converging. And molecular orbitals are a little bit strange. They, you know, the very high-energy ones don't exactly mean something. So they sort of, they're a little bit less refinable, or a little bit less systematic in the way they converge than plane waves, which, you know, converge in really a very systematic way. So anyways, OK, I've hopefully convinced you plane waves aren't like the end of the world, and I'm going to sort of move on. So how are we going to get the n squared terms with plane waves? Well, what happens when we Fourier transform the plane wave basis? A lot of you are probably thinking you're going to get a grid. What I'm thinking of specifically here is you can take the Fourier transform in the mode operators and just plug that into the plane wave Hamiltonian. Well, I mean, you don't get a grid. And the reason why is because the discrete Fourier transform is a unitary transformation. And I'm transforming just a finite number of plane waves here. And so you can't go from a finite number of continuous functions unitary transformed into something not continuous. So what I've shown here is it's actually the square of the basis functions that you do get. It's sort of like each color is a different one. So it's sort of like plane waves trying very hard to be a grid. And they're a little bit different, though. I mean, the key thing is that they do overlap. And because they overlap, that's sort of what ends up admitting this Galeric Indiscrimination part is not pathological in them. But anyways, if you just plug this into the plane wave Hamiltonian, you do get a Hamiltonian of this form. It does diagonalize the periodized Coulomb operator anyways, which is maybe not so surprising, but it's a very nice feature. It's something that's very hard to take advantage of classically because classically, if you work in this basis, the initial state is very much not packed, which is usually what you're looking for. But quantum mechanically, we don't really care about that. We care about how many terms there are. So this is sort of a nice feature. And there's also a nice form, again, for the coefficients here. And there's nice form for these, but I'm not going to bother. Something I'm not really going to spend a lot of time talking about is that as many of you know, you need to map the fermion operators to qubit operators somehow. And when you do that, say, using the Jordan-Wigner transformation, you tend to get qubit Hamiltonians that look like this. So this came from the kinetic operator, and it's an n local term. And there's n squared of them. And it's still better than this end of the fourth Hamiltonian, but it doesn't look like a great thing to implement on your planar lattice or a linear array of qubits. But in sort of our first paper on this, we showed that there was sort of a straightforward way of implementing a whole trotter step of this Hamiltonian in depth of n on a planar lattice, which can, of course, be used for a variational algorithm, can be used for an error correction, trotterized phase estimation, all that. And the way that we did this was we recognized that in the plane wave basis, the kinetic operator, that's at this part here that looks terrible, is diagonal. And you can simulate it very easily. And then you can apply this Fourier transform in n depth on a planar lattice. This isn't the quantum Fourier transform. It's not related at all, actually. It's sort of more like a fast Fourier transform. And people have looked at this sort of fermionic fast Fourier transform before. It's like the fast Fourier transform, but there's some extra signs because they're fermions. But we showed you could do it on a planar lattice, which wasn't, I mean, it was somewhat obvious, actually. But so anyways, you can do that in n depth. And then the other part, the potential, becomes diagonal. And even on a planar lattice, you could then simulate that in n depth. But it turns out we could do even better than this. And we showed this in an even more recent paper. And so not only can you do better in terms of the total number of gates, but it turns out this algorithm works on a linear array, which was quite surprising. So this is the Hamiltonian again. And I'm going to show you how to do a trial step of this on a linear array in gate depth n. So the first thing I'm going to talk about is how one might implement this term here. So we have n squared of these ZPQ terms. And so basically what you'd need to do is you would need to put each qubit adjacent to every other qubit at one point so that you can implement this term, which couples the two qubits. And so you do this using the swap network. And there's sort of a straightforward swap network that does this. So if you imagine, this is my line of five qubits. And these are the different layers of gates. These black things are going to be, imagine it's a swap gate. You see that using the swap network that looks like this, it's sort of fully parallelized except for one qubit. And you just shift it back and forth. If you do this in n layers, every qubit sort of goes past every qubit and is adjacent at least once. So that's very simple. And that allows you to implement this term in a depth of exactly n. But that didn't solve the hard part, because the hard part is what are we going to do about this term that is n local and has n squared terms. So to deal with that, we're going to use what's called the fermionic swap. So again, this has appeared in some prior simulation literature but not really in the context of the algorithm we're outlining here. But the idea is that there is some gate which it essentially changes the Jordan-Wigner encoding. So it changes the canonical order of the fermion. And so it's actually swapping orbitals, which is different than swapping qubits. Because if you look at this thing up here, I mean you can swap qubits all you want. And qubit 7 and qubit 11 might be next to each other. But you're still going to have a string of like 4 z's somewhere else. And so it's different than just swapping the orbitals. Because if the orbitals are adjacent, then this term is just xx plus yy. And it turns out that this fermionic swap gate, it's in general not a local gate. But it is a local gate if applied between adjacent orbitals. So both the kinetic operator and the fermionic swap are two local qubit operators under the Jordan-Wigner transformation if applied to neighboring qubits. And because of that, what you can basically do is imagine that instead of this black thing being a swap, imagine it being a fermionic swap. And so now imagine the qubit's not moving. And instead the fermions, the orbital, not the fermions, the orbitals are going to be moving. And this gate is different than a swap gate. It's something else. But anyways, at doing the same network, it will place every orbital adjacent exactly once, at which point you can implement this zp, zq term. And you can also implement this term, which is just xp, xq. And they're adjacent at that point. So doing this, you could implement all the terms in the Hamiltonian in a very straightforward fashion. And you might imagine doing all of these operations at once, meaning you could do the diagonal part of the potential. You can do the kinetic part. And you can do the fermionic swap all at once. And you write that down in a single gate that would look like this. You could say that this thing was in your gate library. And if you were to say that, then you would conclude that you need exactly n squared over 2 gates of exactly n depth to simulate a charge step of a whole Hamiltonian. Of course, I should mention that, say that you don't have this gate in your library. Why wouldn't you? But let's say you don't. You can, of course, just decompose this into any three entangling gates. So this can be decomposed into a series of three CZs in some rotations or something like this. But I think that this is sort of remarkable because for an explicit trotter step, this appears optimal in some sense, even for an arbitrary connectivity device. Because we have a Hamiltonian that has n choose two interactions and n choose two parameters. And yet, we're able to implement a trotter step of this whole Hamiltonian using exactly n choose two gates on a linear array. And it's fully parallel, so it's also sort of optimal depth in that sense. So this means that even if you gave me an arbitrarily connected quantum computer, I could do any two local gate. I would just draw a line through your device and simulate my algorithm on a line because I don't know how to beat it. Which is kind of surprising. It's not what we would have expected at all. I don't really have time to talk about it, or at least I didn't think I did. I'm going pretty fast here. But we can also implement an arbitrary single particle basis change in depth of just n over two. I should mention that this result borrows heavily from some previous Microsoft work from Hastings, Wecker, Bella Bauer, Matthias Troyer, and Nathan. Which showed that this procedure for preparing arbitrary slater determinants using given rotations. So it's based on that, except for we specialized it to a linear array and also showed you could take advantages of some things like spin symmetry and particle number to reduce the number of gates required. There's also a very nice paper out recently from, it was a collaboration between people at Michigan, NASA, and Google. I was not a part of this one actually. But this was basically looking at doing the same algorithm for the Hubbard model. So actually if you just take the exact algorithm I outlined, we've shown in this paper that it allows you to do charge steps with the Hubbard model in square root n depth on a linear array, which I think that's good news. But also this newer paper shows techniques for preparing the initial states of the Hubbard model that you would probably be interested in for say D-Wave superconductivity, which would, you need a different state preparation procedure and they show how to do that too. So yeah, that all seems good. So I've put together this kind of crazy table here which is the history of these algorithms according to me I guess. So these are all sort of advances in quantum algorithms for quantum chemistry. So I have the year here, the archive number. This is the representation like Jordan Wigner Gaussians, Bravik Tive Gaussians, C.I. Matrix Gaussians. This is the algorithm. So some of these algorithms are variational algorithms and variational algorithms, they are heuristics. Let's not forget that. Because of that I've listed their ultimate cost as a lower bound. You know, because it's just sort of what they cost. Maybe that's sort of an abusive notation. The other algorithms here are for the phase estimation, the phase estimation based algorithm which then requires some time evolution that you either do with typically Trotter or Taylor or now signal processing. And I've suppressed logarithmic factors to the most part here. There's the layout of the chip here. So all these algorithms, variational or the time evolution ones, they tend to have the structure that there's some primitive like a Trotter step or the selective E in the signal processing or something. And then you repeat that a certain number of times. And I've broken them up because the primitive depth tends to reflect improvements in the implementation of the algorithm. I'm giving datas here because it's sort of like, well you know the cost of your algorithm. Whereas the repetitions that has to do with like tighter bounds typically. So you can see the cost has come down significantly since people started getting serious about quantifying the costs here. It sort of, the starting gun was sort of fired by the Microsoft guys actually, I think at this conference four years ago essentially. And they showed these terribly pessimistic results that this is gonna take 300 million years because it's end of the 10th scaling. And just sort of immediately after other groups including Microsoft and the Harvard people and a number of other groups just started hacking away at this and it's been coming down. And so as Garnet was talking about advances in classical algorithms for electronic structure, the same thing is sort of going on here which I think is good news. And we're starting to get down quite low here now. And you can make a variational algorithm of course out of these trotter steps I described. And that would have end depth on a linear array. So that's really, yeah so that seems good. Nathan is gonna show some unpublished data from a collaboration we have. So Nathan's speaking next. Which is looking at using these algorithms I described for a particular problem we're interested in and it looks like the scaling is even improved over this. So yeah, I think that's exciting. I need to quickly plug my, or so we, it was started at Google but it's really not a Google project at this point. There's 23 collaborators. And so it's basically an electronic structure package for quantum computers. So if you want to compile algorithms for chemistry or Hubbard models or any of these things to a quantum computer, you should check out www.openfermeon.org. It's all open source, Apache 2. There's like I say a lot of contributors. It's built in a platform agnostic way and it will remain that way. Meaning that we're not going to wet it to any sort of particular circuit simulator or hardware compiler or something because we want this to be sort of an open platform that can work with IBM QuizKit or with Liquid or with Project Q or whatever you have. And so I think this really sort of simplifies what's required to obtain and study these quantum chemistry algorithms because now you can just sort of, you put in the molecular geometry and you choose a basis set and you choose Bravik Atayev or whatever and it will compute the molecular orbitals, do the integrals, give you the Hamiltonian, you can do all sorts of things with it. So check it out. All right, so what's next? Well, modest goal here, but we would like to work towards trying to implement a variational algorithm like this on hardware in the classical intractable regime. This is no small order, but I think it's something worth working towards and there's really a large variety of challenges that one needs to overcome here. So one thing I didn't really talk about at all is I think there's a significant need for better strategies for measuring the energy and then for feeding it back and doing the optimization. So actually, IBM's recent experiment, I think sort of highlighted this and they have produced some nice solutions to say the optimization and things like this, but it's something that I think we can get a lot better at too. I also think that there's still room for better variational algorithms. We really need to think about how we can use this resource we have that creates very complex states in order to model the complex states that we actually have. And I think that, for instance, that there are ideas that's following the development of the classical methods, for instance, Intenser Networks that might be able to teach us things about the structure of the ground states we're looking for, which we could then use in our circuits. So I think that's encouraging. There's a question of what systems you might wanna look at first. Maybe there's sort of a quantum supremacy and electronic structure, like what's the first electronic structure problem that you could solve on a quantum computer that you probably wouldn't be able to solve otherwise. And in one of our papers, the first paper I talked about, we spent a lot of time talking about why I think Jellium, which is an awesome name, the uniform electron gas seems like an especially attractive target. So there's a lot of nice things about Jellium, but one of them is that it is a widely sort of practice benchmark for classical methods, especially in the quantum Monte Carlo community, there are thousands of papers that sort of measure themselves about how well they do on this specific problem. Now Jellium, it's literally just a molecule without the nuclei. So it's just like the essence of the electronic structure problem. And so it's very simple in a way, but it's very classically intractable in some regimes. And even its size is as small as, I mean, 100, getting the correct answer within the basis is even with enough accuracy to resolve the difference between phases is classically intractable. Also for Jellium, it turns out the plane wave basis is optimal because the Hartree-Fox solution to Jellium is plane waves, which means that you can't complain about the basis I'm using because it's the optimal basis. So that's nice. It's also a rather important system. So it has, you know, Jellium is in two dimensions. One of the systems where you see the fractional quantum Hall effect in where it's studied. But also it's sort of tied to the history of density functional theory in an important way because sort of the universal functional doesn't depend on the nuclei and Jellium is just the molecule without the nuclei. So the universal functional is just sort of like getting the energy of Jellium in a sense. But also the first density functional and the most widely used one still probably, the LDA functional, which is the one that scales to the largest system sizes today, is based on approximations to the energy of Jellium. So if you can calculate the energy to Jellium better, you could potentially improve the LDA functional. You know, I don't, LDA is already pretty not working well for other reasons. So I don't know if that's really gonna change things, but it's a cool connection. Also I think that these improvements are not just relevant in the context of near term devices. So I didn't speak much about what we did in terms of doing sort of new LCU based approaches for these algorithms and so forth. But I think that they're starting to make air correction of chemistry look a lot more plausible. So something that I'm really a fan of people doing is counting teagates just as Andrew Childs was presenting to us today, because that really helps us figure out what the cost is gonna be with an air correction. And from talking a lot with Austin Fowler in my group, at least his perspective is if we have less than a billion teagates and we have something on the order of low hundreds of logical qubits required, then you can potentially do the state distillation in series, meaning you just have only one tea factory. And if you have less than a billion teagates for the algorithm, then that would be feasible. And the long story short is this means that you could solve these problems with, given the air rates that we have today, if we made our device, it's planar and had more qubits, a million physical qubits, that we could solve these problems, which doesn't seem so ridiculous to me anyways. So another thing is I think that there's more work to be done here with connections between the basis and the algorithm still. It's sort of this boring topic unless you're really into the field basis, like it's not that exciting, but it is that exciting. So we should develop, so for instance, classically recently a lot of methods have improved by looking at the use of correlated basis sets. So instead of using orbitals, well you'll still use orbitals, but in addition to using orbitals, which are single particle basis functions, you might use geminols, which are anti-symmetrized two particle basis functions. So it would be like adding a hardcore boson to the system. They can explicitly resolve the electron electron cusp and they can change that algebraic convergence of the basis set error, which I think, you know, that could be nice. Also if you really offended by these basis functions and you're interested in error correction, you can just go to first quantization, where the number of qubits is the number of particles times the log of the number of basis functions. And so you can, 20 qubits gets you a million plane waves and I don't think anyone's gonna complain about that. So that was the talk. You can take your revenge for my aggressive questions now. Okay, thanks, Ryan, for a nice talk. We have some time for questions. Let's start with Jens. So my question starts off at the very end of your talk, which is like, why is there so much emphasis on Gaussian orbitals or plane wave functions? Like given that in classical methods, like tensor network methods or DFT, like these highly curated sets of orbitals like key to the functioning and not Gaussian orbitals? So I would answer this by saying that in fact that the electronic structure community is essentially divided into two camps and for whatever reason they tend not to talk so much and you may only sort of be more familiar with one of these camps, but I mean about half of theoretical or computational chemists do use plane waves as their primary basis. And this is especially if you're looking at materials. Now for single molecules, very often Gaussians are preferred and there's a number of reasons for this. I mean, so one thing is that when you don't have, when you're not worried about electron correlation and if you're using say DFT or mean field methods or even some methods beyond that, there is no electron electron cusp because it's a single determinant method. And if there's no electron electron cusp, the only thing to worry about is the nuclear cusp and the Gaussians are very good at resolving that. But if you're really interested in getting chemical accuracy in the correlation energy, then this advantage between Gaussians and plane waves is less clear. Of course, for systems of reduced periodicity, you do need extra work for plane waves and you know, but you know, let's remember though that looking at materials is a very important problem too and plane waves are the standard there. Further questions? I see one in the back, Kristen. Yeah, sorry. So I have both a question and a comment. So I have to make a comment on one of the earlier slides where you commented on the stuff we did here. Yeah, yeah. And sorry. And so the thing is I agree with you that if you were to do just a hardly far calculation, the energies would be better on those curves. But the reason, I mean, that leads to questions of why do you do experiments like that at all? It's like that's six qubits, that's something you can do on your watch these days, right? And so what is it that you want to gain from them? And I agree that there's ways you can tweak the experiment to make them look better and get the energy down lower. But I think the main point of those experiments is actually to really understand what is it that goes wrong in the actual, in the actual. So this is why we didn't care at all about how really good we are with the actual curve, but we cared more about how good we are with the noise model of our model. So I guess the response to that, though, is that what the noise is going to do to the system depends on your onsots. And your onsots, in my mind, is a combination of the initial state you start in and the circuit you apply. And also, to some extent, even the Hamiltonian you're measuring. And all these things are different if, for instance, you start in the Hartree-Fox state. And certainly, were you to start in that basis, you'll get a better result. Because just the state you initialize in before you apply an entangling gate will have lower energy. I agree that it will have a lower energy. But I mean, the noise process is still the same noise process as T1, T2, D4, and that's what we mostly cared about in the entire process. But I'm seeing that that will have, I mean, how that affects different circuits it depends on the circuit to some extent. You'll see different errors if you use a different circuit. So to the extent that you're interested in studying that. Well, I also want to mention, though, I actually, I really like that experiment. It's really important to actually do these things because there's a lot of complexity in things like the outer loop and using more parameters and better representations, which you guys used in terms of the tapering qubits and this sort of thing. So I didn't really mean to be too critical on that. And then I have another question. So you're switching between the basis. Do you see that like, do you plan to use that in the context of a variational algorithm? Or do you plan to use that in the context of time evolution? So we're not switching between the basis sets. Or what do you mean, are you talking about the T and V, the split operator? Yes, exactly. Yeah, so that was in our first paper on the topic. That's what we came up with. But then we had the second paper where we realized we could do this fermionic swap network thing and we don't do that anymore. We do them all at once. But you plan to use that more in the context of time evolution or variational algorithms? Oh, both. I mean, certainly it's now the lowest depth structured variational thing I can think of. So it's certainly what I'm interested in for variational algorithms. But also Nathan will tell you some numerics for using that in the context of Trotter. OK, so we're out of time. So we have a coffee break now and then we'll reconvene at 4 o'clock with Nathan Wiebe. So let's thank our speakers again.