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0:00
[HOST] --for the fact that Brian was strong-armed into giving this talk today. I sat with him at breakfast at a quantum school in Jerusalem in December, and we were talking, and it seemed like he was the perfect person to sort of give this introductory talk on, to people like me, the com-- computer science audience on sort of the basics of the problems that they look at in quantum algorithms for chemistry. Um, somehow when you just said chemistry, it goes over my head,
0:27
so I'm really looking forward to this talk. I'm really looking forward to learning a lot. And thank you, Brian, for, um-- [BRYAN OGORMAN] Yeah. [HOST] --taking the time to put it together. Yeah. Thank you. [BRYAN OGORMAN] Yeah, so thank you all for being here. You know, thanks to the organizers for what will be a very wonderful workshop. So Yeah, So let me say, I am not a quantum chemist. Um, but, you know, I, I've worked with qu-- cap-- quantum chemists for, for quite a long time on
0:49
applying quantum computation to quantum chemistry. And so I have a sense of, you know, what the gap is in language between, you know, quantum chemistry and theoretical computer science. And so you may have heard that quantum chemistry is one of the most promising applications of quantum computation. And my goal of this talk is to explain, you know, what that means from a computer science perspective so what the main problems of interest are and how and why
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quantum computers may be useful in helping us solve them.
1:22
Uh, so I'll start with you know, a, a reasonable explanation of, of what chemical Hamiltonians are, wha-- what the problem is that we care most about solving. I'll very briefly touch on classical approaches to solving this, and then I'll spend some time talking about how we can use quantum computers to, [AUDIENCE MEMBER] To solve these problems.
1:44
[BRYAN OGORMAN] So I'll start with, you know, the, the very simple model of the world that is the starting point of our quantum chemistry simulations and this is that we have a set of nuclei somewhere in space surrounded by a cloud of electrons.
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So we have these classical nuclei that are these point charges somewhere in space, you know, these black dots in this cartoon and we have these quantum electrons that are in this cloud around it. And of course, these nuclei are really quantum mechanical particles, but they're so much more massive than the electrons that it's a reasonable, very good approximation in this context to model them as classical. It's just these point charges, and this is known as the Born-Oppenheimer approximation.
2:30
But the same is not true of the electrons. You know, there is something about the scale of the system such that, you know, we cannot even qualitatively describe what's going on without modeling these electrons as quantum mechanical objects.
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And of course, this picture, you know, neglects, you know, relativistic effects and others but for the purposes of predicting chemical reactions, this is the starting point of the model.
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And so if we imagine that we fix the positions of the nuclei in space, you know, the electrons will rearrange themselves and we'll get some effective energy.
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And we can imagine doing this for every possible positioning of the nuclei in space, and if we do this, we'll get an energy function that takes these nuclear positions and gives us an energy, and in the space of these nuclear coordinates, we'll get this energy landscape. And this energy landscape is of interest to us because it turns out that molecules are simply local minima in this energy landscape and chemical reactions are just movements from one of these local minima to another.
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And we have, you know, complex chemical reaction pathways are, you know, pathways through this energy landscape.
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And so, you know, as a very simple example, consider a hydrogen molecule. We just have these two nuclei, two protons, and up to rotational symmetry all we care about is the distance between these two atoms, and we get this very simple one-dimensional energy landscape where if they're very far apart, you know, they don't really see each other. When they're, you know, on top of each other, we get these very strong repulsion but somewhere in between there's this sweet spot where there's a local
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minimum in the energy landscape. And indeed, you know, this is why it makes sense to refer to a thing such as a hydrogen molecule. This is the state that the system is in most likely, you know, up to thermal fluctuations.
4:26
And so, you know, even if we could compute this energy function as a function of the nuclear coordinates, there would still be this hard problem of exploring this landscape and finding out where the minima are and, and, you know, pathways between them But that's not the problem that I'm going to be focused on in this talk. Uh, the, the problem that I'll be focusing on in this talk is just the evaluation of this energy function. So I'm given a fixed position of the nuclei in
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space, and the question is, what is the energy of the electronic states as a function of these nuclear coordinates? So there's one part of this energy that just comes from the interaction of the nuclei amongst themselves but remember we're modeling these nuclei as these classical point particles. There's this very simple Coulomb interaction between them. This is trivial to calculate. The harder part is this electronic contribution to the energy and it's harder because, you know,
5:18
these electrons are in some quantum mechanical states. And so the main goal of quantum chemistry, the thing I'll spend, you know, almost all of the rest of the talk talking about is finding the ground states of this electronic Hamiltonian. So we fix the positions of the nuclei. They exert some fo-- um, they exert some forces on the electrons, and we want to find the lowest energy states of these electrons subject to this Hamiltonian. And it'll turn out that this Hamiltonian has a,
5:45
a nice form once we've sp-- spent some time developing the formulas and explaining, you know, where these coefficients come from and, and what these funny operators are because they're not the, you know, normal qubit operators that, that we're used to seeing. To start with a very simple example, you know, consider the state of the single electron. It can be described by this wave function in real space, in R3. And of course, it's a quantum state, so it's normalized.
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But this wave function is this infinite dimensional object. But at the end of the day, we want to compute something. We want to represent it on our computer, either classical or quantum mechanically, and so we need to discretize. We need to somehow make our representation of the system finite. And we do that by introducing this finite basis of the wave function. Uh, and in this context, I'll call these one-electron basis functions orbitals. And I'll use this word, you know,
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many times throughout the talk. You know, orbitals are just these one-electron basis functions. And so, you know, given some wave function, we can at least approximately write it in this basis of one-electron basis functions. And if we do that, you know, we can, you know, rewrite this more physics-looking expression as something that we're familiar, right? This is just a big N level qubit.
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But note that this approximation could be good or it could not be so good. And so now, we've introduced what is our first non-negligible approximation, where we've taken this infinite dimensional space and approximated it by a finite dimensional one. And so the difference between these two things is called the basis set error. So it's the difference between, you know, the true ground state energy, the ground state of this full infinite dimensional Hamiltonian, and the ground--the best,
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the lowest energy state within the space spanned by our finite number of basis functions. And this choice of basis functions is not arbitrary. You know, depending on the system, you know, we'll have some sense of what the ground state should look like, at least qualitatively, and we'll want to choose our basis function so that we can capture as much of that ground state using as small a number of these basis functions as possible. And a huge part of classical computational chemistry
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is finding good basis functions, so that we can, as concisely as possible, represent the ground state of systems of interest.
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So now, moving on to two electrons. So now, we have a wave function that's a function of just the position of these two things in space. And again, we introduce this one-electron finite basis. We use the same basis for each electron. And so the wave function of these two electrons, we can write, again, at least approximately, in a basis of tensor products of these one-electron integrals, one-electron basis functions.
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And so again, you know, we can rewrite this physics-y looking expression in something that's familiar to us, which is, you know, two end level qubits.
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But the problem now is that, you know, we're talking about a physical system. And in fact, these electrons are identical. They're completely indistinguishable. It doesn't make sense to say that the first electron is in some states and the second electron is in some other state. So, and we need to, you know, enforce this into the wave function.
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And, you know, mathematically, this means that the wave function must be either symmetric or antisymmetric when we permute the particles.
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And so at this level of the theory, you know, we'll just need to take it as a rule. You know, it can be derived at deeper levels. But at this point, it's just an assumption that we're going to make that electrons are what's called fermions, which by definition means that they're the type of particle for which the wave function must be antisymmetric,
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as opposed to symmetric.
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And, you know, operationally, what that means is that we need to restrict our attention to the minus one eigen space of the swap operator, so that when we swap the state of the two qubits, we pick up the sign, but otherwise, leave the state unchanged.
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And so, you know, we saw that we can write our system as, you know, the general states of two qubits. But now, we're gonna look at a smaller state spanned by, you know, antisymmetric combinations of unique indices. And, you know, I emphasize at this point that, you know, these pairs of indices are unique, because this antisymmetrization property implies that no more than one electron can be in any particular state. Because when we apply the swap operator, you know,
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if there were two things in the same state, we would get that the coefficient is equal to minus itself, and so it has to be zero. And this is known as the Pauli exclusion principle [AUDIENCE MEMBER] I-- in chemistry.
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[BRYAN OGORMAN] And so this suggests that maybe there's a different way of representing our state. Instead of talking about the states of individual particles, we'll just talk about the occupations of states. So we can start to index our states by these bitstrings with, in this case, exactly two ones in them, which just say which orbitals have electrons in them. But this representation, you know, directly captures the fact that these are indistinguishable particles.
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You know, it doesn't say which electron is in which state. It just says these are states and they have electrons in them.
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And so as a very concrete example, you know, suppose we have two electrons and four orbitals. You know, so on the one hand, in this particle representation, we have this basis of antisymmetric combinations of unique indices.
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But it corresponds in this occupation representation to bitstrings with exactly two zeros. And so, you know, for two electrons and four orbitals, we get these 4 choose 2 or 6 basis elements of this space. And this is just a relabeling of things, right?
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We saw that we have to have this antisymmetrization, and so it makes sense to, you know, start to throw away information about which electron is which and have our notation, you know, more concisely capture the nature of the system that we're working with.
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And now, we can start to generalize this to an arbitrary number of electrons. So, you know, at a first pass, we have just a set of many large dimensional qubits, but again, we need to restrict ourselves to this antisymmetric subspace. Uh, and so for a general set of, of n electrons, this means that for every pair of qudits, we need to restrict our attention to the minus 1 Ogden space of the swap operator for, for all of these qudits. And so again, you know, we're restricting our attention
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to this antisymmetrized space, which we can, you know, iterate over by looking at all sets of unique indices and all permutations of these indices with the addition of a plus or minus 1 depending on the parody of the permutation.
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And this should strike you as very familiar because this is exactly the form that determinants take.
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And indeed, you know, there's this n-by-n matrix that I can write down where the columns correspond to particles and the rows correspond to orbitals such that, you know, these basis states are exactly determinants of matrices of that form.
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But again, you know, this is a little complicated. It obscures what's going on. And so we have this alternative occupation representation in terms of bit strings with exactly some number of ones in them which say, you know, which orbitals are occupied by some electron.
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And so, you know, given this occupation representation, you know, I, I, I've explained what the states represent, but now we need to start operating within this space. And so we define what's called an annihilation operator. And this annihilation operator takes bit strings to bit strings, or, you know, states indexed by bit strings to states indexed by bit strings, and there's an annihilation operator for every orbital. And so if we, you know, act with an annihilation
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operator on a state that has an electron in that particular orbital represented by this one, it knocks down the occupation number to zero. But it also introduces this antisymmetrization factor that depends on the occupation of some other set of orbitals. And so note that this occupation representation, you know, by construction represents the indistinguishability of these particles, but the fact that it's antisymmetric rather than symmetric, it comes into the way that we operate on these states
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and keep track of the relative sign between these basis states.
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And, you know, on the other hand, if we act with this annihilation operator on a state that doesn't have a particle in the corresponding orbital the operator completely extinguishes the state because there's no particle in that orbital to remove. And then we can also look at the Hermitian conjugates of the annihilation operator called the creation operator, which does the opposite. So if we act on it, with it on a state that doesn't have an electron in that orbital,
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it bumps up the occupation number to one, again with this antisymmetrization factor. But if we act with a creation operator on a state that does have an electron in that orbital, then it completely extinguishes that state. And this, again, comes from the fact that these represent antisymmetrized state, and there can be no more than one particle in each orbital.
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And so we're, we can start to combine these operators, and, you know the first thing we can do is create what's called a number operator which just kicks out the number of particles in each orbital which will just be zero or 1. And we can also look at other combinations of these things. So in particular, if we take a creation operator and annihilation operator and we look at the anticommutator, we get 1. Yeah? [AUDIENCE MEMBER] So the way you wrote these it looks like
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the creation and annihilation operators are not local with respect to the orbitals. Is that the case? [BRYAN OGORMAN] Yeah. Yeah yeah. And, and I'll get to that later because that, that introduces some complication when we want to represent these states in a state of qubits. So, you know, this looks like a set of qubits, right? Because I have these states that I'm indexing by you know, bit strings, but these are not qubits. These are fermionic states. And so we have these, you're right,
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what look like non-local operators and that will require addressing when we want to represent these as qubits and manipulate them in, in some local way. Non-local in the spatial sense [MODERATOR] Because-- [BRYAN OGORMAN] Local, well, it's non-local in the, the general sense and the number of things that it acts on non-trivially. [LECTURER] Oh, okay. [BRYAN OGORMAN] Which, you know, implies that it's non-local in the geometric sense as [LECTURER] Well. Right right. [BRYAN OGORMAN] Okay, yeah.
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But we'll return to that very explicitly. I'll give some constructions about how we address that.
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Uh, but that is one of the, you know, it's a good question because it gets at one of the main differences between these fermionic Hamiltonians and, and general qudit Hamiltonians. So we have a Hamiltonian that looks local, right? I wrote it as the sum of these things over some small number of indices, but these operators are non-local in some sense. [AUDIENCE MEMBER] Oh, and sorry, one other thing. The order, so the, P and Q are vectors, right? <|agent|><|en|> P and
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[BRYAN OGORMAN] Q, no, they're just single numbers between 1 and big N. [AUDIENCE MEMBER] Oh, right. Sure.
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Okay. [DAYWAN] Yeah, Mark? Any intuition for why there's that ordering that when you annihilate at site P, that it only cares about the occupations-- So [BRYAN OGORMAN] This is-- Yeah yeah. So, this is completely a convention. So, you know, we have these bit strings that represent, you know, occupation vectors let's say, you know, which orbitals have electrons in them.
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But what's important is that we keep track of-- If we look back at, you know, the correspondence between these bit strings and the antisymmetrized product, there's this, you know, sign in front of the permutations that is the parody of that permutation. And really, if you think about it, you know, what matters is not the parody of any particular permutation, but the relative parody of different permutations. And so the fact that this sign is only a function of
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some occupation numbers and not others, is really just a convention that, you know, we choose some ordering of the orbital so that we can consistently keep track of the relative sign of these things.
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[DAYWAN] So there's an implied mapping between orbitals and the-- [BRYAN OGORMAN] Yeah, yeah. So it implicitly when I say that we have this bit string in this occupation basis, there's an ordering of things, right? So I'm saying that, you know, the first orbital is the first bit, the second orbital is the second bit, and you know, the ordering is completely arbitrary as long as I'm consistent.
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Okay. Any other questions?
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Yeah. So okay, so we defined the-- this number operator and then we can look at this anti-commutator and, you know, it turns out that it's one because no matter what the occupation number is in the Pth orbital, you know, when we act on it by this anti-symmetric or this anti-commutator, one of these terms will be zero and the other term will return the state unchanged. And so no matter what we state we act on it with, we get one, or the state returned.
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And so now we can look at combinations of annihilation operators corresponding to different orbitals.
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And again, we see that the anti-symmetry is manifest in the way these operators interact with each other, so that if we switch the ordering of these operators we get a minus sign. But otherwise things are unchanged. Um, and, you know, in a sense, the fact that the minus sign is here rather than here comes from this-- I-- I'm just picking some ordering and, you know, in whatever indexing scheme I've chosen, you know, say P is less than Q or, or vice versa.
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And as long as I'm consistent, things will work out. But in some sense there's an arbitrary choice I make to begin with. And so overall we can, you know, very succinctly describe the relationship between these operators in what's called this fermionic algebra. So this very succinct set of anti-commutation relationships. And it'll be also useful later on to think of this identification between these bit strings these occupation vectors and the set of creation operators that when we apply to
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the vacuum state, the state with no particles in it, creates this occupation vector. And so, you know, if these things were to commute, you know, we can think of this as just a multi-nomial polynomial, or sorry, multi-linear polynomial. But they don't commute, so we need to be careful about the sign. But it'll be useful later when we talk about, you know, changing basis and stuff to have this picture in mind.
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So now we can get back to the Hamiltonian, th-- that we care about. Um, so the laws of physics tell us that if we have these, these point charges and we have this set of electrons, this is what the Hamiltonian is. This we have no control over. And the, the Hamiltonian has, has three different parts. So one is the kinetic energy of the electrons. Um, the second is the, the Coulomb or electromagnetic interaction between the electrons and the nuclei. And the last is the, the Coulomb interaction
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between pairs of electrons.
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In an instance of the problem that we're interested in is defined by three very simple parameters. We have, you know, the set of nuclear positions, you know, this set of vectors in R3.
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We have the nuclear charges, which just counts how many protons are in each nucleus. You know, what is the charge of each nucleus and how many electrons we have. And note that, you know, all of the electrons are identical, so the only thing that matters is how many there are. There's no sense of first electron, second electron, and [LECTURER] So on.
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[BRYAN OGORMAN] But then again, you know, this is defined in this infinite dimensional space, and so we'll need to pick some finite dimensional basis in which we represent this Hamiltonian. And so we do that And, you know, which orbitals we choose can make a big difference, but we choose one And then we can calculate the, the representation of the Hamiltonian in this basis. We get these coefficients, these tensors whose elements come ultimately from, you know, integrals over these energy
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functions of the individual one electron basis functions.
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And so we get back to the Hamiltonian that, that I started with, and said that, you know, I would explain why it has this form and, and where the-- these coefficients come from. So this is representing the Hamiltonian, what's called the occupation basis. So we have this one body term that corresponds to things having to do with a single electron, the kinetic energy and its interaction with the nuclei. Then we have this two body term that corresponds to
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things having to do with two electrons, namely the, the Coulomb interaction between them.
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And, you know, this is very nice. It looks local in some sense, but as we talked about earlier, it's non-local in another sense in a way that we'll need to address.
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And importantly, it's anti-symmetrized by construction.
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But note that it's still QMA hard, right? So if we were to look in the particle representation that is not inherently anti-symmetrized, we would say, you know, we're trying to find the lowest energy state subject to this anti-symmetrization condition, and so we're restricting our attention to a smaller part of the space. And it's important to note that that's without loss of hardness. We can still-- This still captures QMA complete Hamiltonians.
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There's a very straightforward mapping reduction from a two local qubit Hamiltonian to a Hamiltonian of this form. And so even though we've anti-symmetrized things, things are still hard, except for the fact that, you know, these coefficients come from these integrals of a very particular form. And so maybe, you know, there's some structure here that we can exploit, at least heuristically. Maybe we can say, you know, oh, we'll, we'll find the ground state, but, you know,
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there's some structure that we can start to get some intuition with and, and develop good approximations for. Another difference from the, you know, generic local Hamiltonians is that w-- normally when we're talking about local Hamiltonians, we imagine we have a large number of particles with a small local dimension. But here generally, and especially for the types of instances that we think quantum computers will be most helpful for, the situation is a little reversed.
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So we have some small number of particles, you know, tens of electrons or something like that. But the number of orbitals or local dimension is much, much higher. It's orders of magnitude higher. And so, you know, this isn't technically inconsistent with our-- our general idea of local Hamiltonians, but it has, you know, a qualitatively different flavor that-- that may use-- require different techniques.
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This is a good place to stop and ask if there are any questions. You know, this is as-- as much as I'll say about, you know, the problem that we're trying to solve.
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[AUDIENCE MEMBER] Yeah? So you mentioned on the previous slide, Or-- Or actually right there that you're interested in the situation where the number of orbitals is much greater than the number of electrons. Mm-hmm. So I mean, what's the motivation? That's essentially s-- So I'm a quantum chemist. Mm-hmm. So like, what's the motivation for kind of focusing in on that? Because typically I think about quantum computers being most useful for strongly correlated systems. Mm-hmm.
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Which is normally around the case where you've got, you know, like half occupancies. So similar number of electrons and orbitals, where this Yeah. [BRYAN OGORMAN] So this is, this is not an absolute statement. It's-- Okay. It generally was, you know, I wrote it down consciously to say that. You know, this is not absolutely the case, but sometimes. [AUDIENCE MEMBER] Yeah, I mean, I think it's, Yeah, It is probably helpful to identify that as that is the common case though.
26:11
[BRYAN OGORMAN] Yeah, yeah. This is not-- this is definitely not always the case. It's-- Yeah. Okay. But often. Often.
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[DAYWAN] Um, [BRYAN OGORMAN] Yeah, and-- and generally, you know, especially if you're a quantum chemist a lot of the things I'm saying might
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grab your attention because, you know, I-- I'm-- I'm simplifying a lot of things and-- and sweeping a lot of exceptions and technicalities under the rug.
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Uh, so now I'll say a very little bit about classical approaches.
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So the starting point of classical methods is often usually what's called Hartree-Fock. So at a very high level, the Hartree-Fock state is the best set of n-electron orbitals.
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Or more precisely, it's the state in which, you know, the best n-electron-- single electron orbitals are occupied. And so to start to get some sense of what this means, suppose that we only had a single electron and we could ignore the, you know, electron-electron interactions, and we ask, "What are the n lowest energy levels of the single electron?" And because there's only one electron, you know, we can easily calculate this and get the n lowest energy states.
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And now suppose that we brought back in all of the other electrons and we populated these n lowest energy single electron states. If we were to do this, you know, once we've reintroduced the electron-electron interaction, this will probably not still be a good approximation of the ground state. You know, somehow this electron-electron interaction is very important to consider. And so what Hartree-Fock does in a sense is it does better by considering, at least on average,
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the effect of these other orbitals. And there's some set of couple of equations that we can iterate through to get this mean field approximation of the best single n single-electron orbitals.
28:07
So a little bit more formally you know, what does it mean to change the basis? So in the particle representation, it just means that we're gonna apply this rotation to each of the qudits individually.
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And in the occupation representation, what we're doing is we're applying this rotation to the creation operators. We're gonna write our creation op-- our new creation operators, these A tildes, as linear combinations of the old ones. And our goal with Hartree-Fock is to find this rotation such that we can find the best single determinant, this single bit string in this new basis that has the lowest energy. And this won't always be, you know, the ground state. In general, it won't be.
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But it's often extremely close. And this is, you know, it's often surprisingly close. And even when it's not, you know, exactly the ground state, it's a very good starting point. And it's why it's Hartree-Fock is the first thing I introduced, because for both classical and quantum computational methods, you know, Hartree-Fock is the starting point. And we have, we get very close to the ground state, and then we want to use, you know, more advanced methods to get a little
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bit closer to the ground state.
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And note that I was talking about this Hartree-Fock state, but, you know, we're optimizing over this basis. And so really, we have this Hartree-Fock basis in which, without loss of generality by convention, we can just say the Hartree-Fock states is, you know, ones followed by zeros, but we have this new basis that we're starting to work in.
29:36
And so the first you know, next step we can start to take from Hartree-Fock is what's called configuration interaction. And so, you know, recall the Hamiltonian, it has this form where there are these one body terms that take a-- an electron out of one orbital and put it in another, and there's this two-body term that takes a pair of electrons out of orbitals and puts them in new ones. And so this induces some sort of metric on the space of bit strings that--
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That's, you know, proportional to the Hamming distance between these bit strings. And so we can start to look at states that are near Hartree-Fock under this metric.
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So, you know, there are some set of orbitals that are occupied in Hartery-Fock and some set that aren't. And so we consider, say, singles where we take a single electron out of the occupied states and put it in an unoccupied one.
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You know, we, we can look at doubles and triples and so on. And, you know, as we increase the number of excitations we're willing to consider away from Hartree-Fock, you know, we're incorporating a, a larger and larger portion of our Hilbert space, and so our approximation of the ground states will be getting in better, but th-- at the cost of getting more and more expensive, because we're, you know keeping track of more and more states. And in the limit, when we consider
30:56
an arbitrary number of excitations away from Hartree-Fock, we get what's called full configuration interaction, which is essentially exactly diagonalizing up to the basis set error. And so, you know, this is the gold standard of quantum chemistry calculations exactly because, you know, we're diagonalizing in essentially the whole space. But of course, you know, this comes with a cost of being extremely expensive, and so we only really do FCI for relatively small sizes.
31:27
And there are other ways of you know, trying to find the important parts of the Hilbert space to search over. So there's a class of methods called adaptive sampling, where we start with some small set of determinants. We look at, you know, a larger set that's nearby, that's connected to it through the Hamiltonian. And so we've increased the size of our representation, and hopefully improved our approximation of the ground state. But we can't keep doing this, right?
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There's only so many determinants that we can keep track of. And so what we do is, we diagonalize in this larger space, and we look at what the most important parts are, and we throw away the rest. And so we return to a new small set of determinants that's hopefully a little better than the one we started with. And we can iterate this on, you know, expand a little bit, optimize, throw away the negligible parts, and you know, repeat this over and over.
32:19
And so empirically, you know, this has been shown to work extremely well. Uh, and it, it's interesting in, you know, to compare with algorithms that have been developed in the context of the ground states of 1D Hamiltonians, where we have these matrix product states that are these bounded size, classical descriptions of an approximation to the ground state. We, you know, use the Hamiltonian to see what is nearby in this--our, our putative onsets. We get this larger and better approximation of
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the ground state but then we need to truncate in some way. And so there's this process of, you know, increasing the size and accuracy of our representation, and truncation back and forth-- truncating back and forth in a way that, you know, hopefully converges. In that case, we can prove it does.
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So that was in some sense, a complete injustice to classical methods. You know, this was a very, very simplistic introduction to what classical computational approaches to quantum chemistry are. And it's important to say that classical computational methods in quantum chemistry have been enormously successful. There are many, you know, very large interesting systems that are very useful that have been solved extremely well.
33:36
But the problem is that there's, you know, Yeah? [AUDIENCE MEMBER] I don't know if I understand correctly, but what you said, like, it seems that the ground states are not very dangled when you were doing this classical method. Is it true? [BRYAN OGORMAN] Yeah, yeah, Yeah. So that's sort of what I was just getting at, is that the systems for which these classical methods work very well are systems for which you can describe the ground state to a very good approximation by some reasonable
34:05
number of determinants. So it's not exactly unentangled, but it's lacking what's called strong correlation. And it's exactly those instances for which that's not a reasonable description that we hope to use quantum computers. Because instead of keeping track of the state of the system as this, you know, classical representation, this list of amplitudes in, in some basis, now we're gonna just create the states on our quantum computer and let, you know,
34:35
Mother Nature keep track of, of the amplitudes for us. [AUDIENCE MEMBER] Could I-- Yeah. --bring on that just a moment, if you-- Yeah. Would an example be, for example, the molybdenum cofactor or the iron cofactor in these enzymes that have strong correlations? Or are those cases that you think are adequately simulated with classical tools? [BRYAN OGORMAN] I don't know. I, I'm not familiar with that, that system. Maybe someone who works with these things in more detail can answer.
35:06
[AUDIENCE MEMBER] Does anyone know? [ALEC] Yeah.
35:09
I, I'll touch on it in the next talk, actually. Fantastic. [BRYAN OGORMAN] Sure. Yeah. So, so another thing I, I should say at this point is that the next talk later this afternoon will be by Alec, who, who will say a lot more detail about classical computational approaches to, to quantum chemistry that you know, are, are, are very successful, except for in some cases, they're not. And he'll say hopefully a lot more detail about, you know, where that line is.
35:40
So very simply, you know, the, what it means to do quantum chemistry on a quantum computer is that we try to create this ground state on our quantum computer. And so there are a few elements that go into doing this. So the first thing is that we need to map the states of the electrons to the states of our qubits. And then we need to do something that, you know, has the effect of preparing the ground state. And we heard a lot about that this morning.
36:05
I won't say much more than to, you know, review the techniques that were explained earlier. Um, but I'll emphasize a little bit about, you know, some special cases for, for quantum chemistry.
36:19
Um, so the, the first thing I'll talk about is this, this mapping between fermionic states and qubit states. So on the one hand, you know, both states are indexed by these bit strings, and so, you know, it may seem like, "What's the problem? Why don't we just map it?" But suppose that we were just to do this identity mapping, where we map a bit string to the same bit string.
36:43
But recall that we have this annihilation operator that looks very non-local, right? It has this parity that depends on the occupation of some other set of orbitals. And so one way, the Jordan-Wigner encoding of mapping this fermionic operator to a qubit operator is like this. We have some string of Zs that acts on the lower-indexed qubits
37:07
with this thing that has the effect of moving from the one state to the zero state-- Ahem. On the pth qubit. And it's exactly this string of Zs that accounts for this-- this parity up here.
37:20
And, you know, I won't go through the details here, but you can show that, you know, if we do the mapping in this way, then these encoded qubit operators obey the fermionic algebra that we showed. They have the same anti-commutation relationships as the fermionic operators that they're mapped from.
37:39
And you'll note that if-- if we do this mapping and we start with this two-body fermionic Hamiltonian, and we encode it i-- as a qubit operator in this way, we get these n-local qubit Hamiltonians which is not good in a certain sense. But it's not as bad as it may seem. So on-- on the one hand, we can still efficiently measure the energy of this thing-- th-- these things, because although it's n-local, these n-local operators are strings of single-qubit or tensor products of single-qubit Paulis.
38:10
Um, and so, you know, in one way of thinking about it is that there's, like, a very simple MPO description of what-- what these Hamiltonians are. But there are some complications when we want to, you know, say, do time evolution or otherwise manipulate the states in a way that's expressed using these operators. And sometimes it turns out for you know, many systems of interest, there are some tricks that we can do where, you know, this overhead sort of goes away.
38:40
But still, it's, you know, it's a complication that needs to be addressed.
38:45
So an alternative way of representing the states is what's called the parity encoding, where, you know, before, you know, we saw that this non-locality comes from the fact that, you know, we have this minus one that depends on the occupation of some other set of fermions or orbitals. And so we can ask, you know, "Okay, why don't we just represent that, you know, and use the--our qubits to represent that?" And the problem is that if we do that, now the, you know,
39:14
the parity is encoded locally, but now the occupation is non-local, and so we've sort of flipped things. You know, whereas before, we had some string of Zs on the left followed by some local operator, now we have some local operator followed by some string of Xs. And so, you know, we've flipped things in a way that doesn't change much of interest. We still get this n-local Hamiltonian when we map from a fermionic operator Hamiltonian to a qubit operator.
39:46
But fortunately, you know, there's some intermediate ways. So there's what's called the Bravyi-Kitaev encoding which maps the fermionic states to these qubit states in a, you know, very clever way. And I won't go into the detail o-- of exactly what's going on here, but the point is that the end result is that when we take our two-body fermionic Hamiltonian and we represent it as a qubit operator using the Bravyi-Kitaev encoding, what we get is a log-local
40:14
Hamiltonian, which is, you know, much improved in terms of starting to manipulate our state using you know, primitives that are expressed in terms of these operators.
40:24
And this was a, you know, very non-exhaustive
40:27
survey of, you know, different approaches for doing these fermionic qubit mappings. And there's, like, a very large and growing body of work that has some very nice generalizations and extensions of these types of ideas, where, you know, some of these are locality-preserving for, you know fermionic systems defined on lattices. And maybe of interest to people here is that some of these encodings have error-corrective properties. So it's not only an encoding of the fermionic states and
40:56
the fermionic algebra but for example, this Majorana loop stabilizer code also has the property that it corrects up to single-qubit errors. And so there's this nice intersection between, you know, this fermionic algebra and quantum error correction that might be a good entry point into quantum chemistry for a quantum computer scientist who's maybe not familiar with physics or stuff like that.
41:22
So having, you know, talked about how we represent the state of our electrons on our quantum computer, now we need to start talking about, you know, how we're going to prepare this ground state. And, you know, I won't tread any new ground here. It'll really just be a review of what we heard this morning. So one way is variationally, right? So we have some parameterized circuit. You know, we have these classical parameters theta that are these little knobs on our
41:47
gates that we can play around with, and try to optimize, and produce the lowest energy state that we can. And it's variational in the sense that, no matter how we set these knobs, we'll always get an upper bound to our ground states. And so we'll never know that we got our ground state for sure, but we will know that we never overshoot it. We never go below the ground state energy.
42:11
And as was mentioned before, how we find these parameters is a big open question. But this is also a place where classical i-- or ideas that we know from classical computation of quantum chemistry can give us a huge headstart. So for example, you know, we can use as an initial state this Hartree-Fock state that we know is already very close to the ground state. So, we're not in what would be the hopeless case of not knowing anything about the system.
42:42
We know that we can get very close to the ground state, and maybe we can variationally just get that much closer to the ground state, and improve the accuracy of our approximation just that much more.
42:53
And so particularly, we can use what's called Unitary Coupled Cluster which is this very chemistry motivated ansatz. So it-- it uses structure from the Hamiltonian. Uh, in particular, you know, the ansatz is the exponential of what's called a cluster operator which has these one and two body terms that look very much like terms of the Hamiltonian, except instead of coming from, you know, the true physics of the problem, these coefficients are parameter-- variational parameters.
43:23
They're exactly the knobs that we can start to tune to try to find as low an energy state as we can. And, you know, maybe we can't do this exactly, but we approximately implement this operator.
43:38
Now another approach that we saw earlier this morning was Quantum Phase Estimation. So, you know, suppose that we were given an eigenstate of the system, and we want to measure the-- we want to estimate the energy of the eigenvalue, or more precisely, the exponent of the eigenvalue. Quantum phase estimation is a very simple quantum circuit that allows us to do this. So if we feed in, you know, some ancilla register and this input's eigenstate, Quantum Phase Estimation
44:04
gives us an estimate to some, you know, finite precision of what this energy is.
44:10
But of course, you know, we don't necessarily have the exact ground state that we want. And so in general, if we feed in an arbitrary state, we get the ground state energy, you know, up to finite precision with probability that's proportional to the overlap with the ground state energy. And this is another case in which the fact that classical methods can get us extremely close to the ground state is of interest, because if we just had an arbitrary state, you know, this would be hopeless.
44:36
We would just be getting random noise from our quantum computer. But if we know that we can start with a ground state, or sorry, an approximate wave function that has very good overlap with the ground state, then there's this hope that, you know, with high probability, this procedure will project us into the actual ground state, and we can estimate the energy thereabout. And there's been some very nice numerical work recently showing that you know, for many systems of interest, we can actually
45:03
make this overlap pretty good, and that this is a very promising approach, you know, down the road when we have computers that can actually implement these things for finding the ground state.
45:17
Again, as we heard before, you know, another way of preparing these ground states is adiabatically, right? So we want to prepare some ground states. Um, so we start in the ground state of some easy-to-prepare initial Hamiltonian. For example, you know, a-- A simple Hamiltonian whose ground state is Hartree-Fock. And we evolve for some time, interpolating between this initial Hamiltonian and the Hamiltonian whose ground state we want to find. And in general, you know, this is complete for
45:43
quantum computation. This could take a very long time. Um, but we-- we can hope that, you know, if we start close enough to the ground states of the Hamiltonian that we care about, you know, maybe-- maybe the general worst-case results a-- aren't necessarily applicable, and that the instances that we care about won't be-- w-- won't be pathological, that-- that the you know, the path between the initial state and the final state will be relatively short. But of course, this is a heuristic,
46:13
and it remains to be seen empirically when we can build these things.
46:19
Um, and the last thing is that, you know, in-- in the previous algorithms, I sort of assumed that as a primitive, we have the ability to do time evolution, you know, given a Hamiltonian, evolve the states under the action of this Hamiltonian.
46:32
And there are many different ways of doing this. You know, a simple way is called the product formula, where, you know, imagine the terms of this local Hamiltonian commuted, then we can just simply apply the product of the exponentials of the individual terms. They don't commute, but it's a reasonable approximation to do it like this, where we evolve a little bit under each term in sequence, and then we repeat over and over again. And so we sort of stroboscopically
46:58
approximate the evolution of this whole Hamiltonian.
47:02
And in the past, there's been a very wide gap between, you know, bounds that we can show on the error of this approximation analytically, and what we saw in practice. That is, you know, empirically, we saw that this is a much better approximation than we can prove it is. And very recently, in the past few weeks, as we said earlier, you know, there's been some very nice work that, you know, takes into account a-- And in very careful ways, the relationship between
47:25
these terms in a way that makes this approach, you know, competitive with other recent advanced algorithms. Um, you know, things like quantum singular value transformation is a-- is beautiful general algorithm, one of whose applications i-- is this task of Hamiltonian simulation. Um, and we'll hear a lot more about that in detail l-- Later in the week from Andresh. [LECTURER] And so [BRYAN OGORMAN] To conclude you know, I-- I've hopefully given the sense that, you know, mainly quantum
47:54
chemistry about-- is-- is about finding the ground states of a specific type of Hamiltonian.
48:00
You know, classical methods can get us, classical computational methods can get us very good approximations using heuristics that are, you know, well-founded physically and, you know, validated empirically. But still there are systems for which they don't get quite as accurate as we would like them to, and these are the instances for which we can start to use quantum computers. Um, and we can try to set up the ground state of the Hamiltonian on our quantum computers.
48:26
And there are these different aspects that go in to do that, you know, interact in subtle ways. So for example, there are basis sets that we can choose that will yield some structure in the Hamiltonian that we could exploit algorithmically-- Um, but we also need to be careful that the basis sets that we choose, you know, are appropriate to the instances that we're looking at. That with some reasonable number of them, we could well approximate the ground state.
48:52
And overall, you know, progress can be made by looking at, you know, trade-offs between these things, and trying to improve parameters and the relationship between them. Uh, a-- and overall, you know, I hopefully I've-- I've given you a sense of why quantum chemistry is a very interesting and useful domain to start to apply these advances that have been made in quantum algorithms on the increasingly larger and better quantum computers that are being built. [MODERATOR] Thanks. (applause)
49:26
Questions for Brian? Yeah.
49:31
[LECTURER] I have a question. So the Hartree-Fock solution, once you have that, can you write out the Hamiltonian for which it's the ground state?
49:42
[MODERATOR] Yeah. [BRYAN OGORMAN] I think that's [MODERATOR] Fair. So you can't just remove all the interaction terms between electrons, right? That will not-- [BRYAN OGORMAN] Yeah. So, I mean, that example-- [MODERATOR] No. [AUDIENCE MEMBER] Yeah. For the Hartree-Fock state, you can, right? That's-- isn't that the whole point? [MODERATOR] You can write out the [AUDIENCE MEMBER] So if I-- if I understood Hartree-Fock correctly, it seems like essentially what the algorithm is doing is so if you,
50:04
so you have that column column interaction, right? Mm-hmm. You can also, that comes from another variational problem, which is just, say, minimalizing the energy of the classical electric field with the charges at particular places. So you could just-- you-- you could essentially think of Hartree-Fock as alternating between solving the classical field to find a single field that then you consider the Hamiltonian that-- that has no electron-electron interaction terms, but it has-- It's a--
50:32
The arbitrary field, and then-- [BRYAN OGORMAN] So, I'm not sure that that's quite right. Yeah. I mean, it's not quite exactly that you're throwing away the electron-electron interaction. You're just incorporating it in some average way. And so it's not quite that there's no interaction between the electrons.
50:54
But maybe a different way of-- [AUDIENCE MEMBER] More concretely, could you write down the the Hartree-Fock state as the exact ground state of something similar to the actual Hamiltonian, but instead you sort of throw away all the electron-electron interaction terms, but you add a term which is electrons to now an arbitrary field, in addition to the ones generated by the nuclei?
51:19
Where,
51:22
[BRYAN OGORMAN] Yeah. So I mean, the Hartree-Fock states in the Hartree-Fock basis is just a you know, a single bit string. And so, of course, there's a very simple Hamiltonian of which this is the ground state.
51:38
But it's not exactly you know, it's, that's not exactly how we find the Hartree-Fock basis to begin with.
51:52
[AUDIENCE MEMBER] Yeah. Yeah. So I mean, maybe another way of saying it, kind of one of the ways that we think about it as chemists is that you do basically seek out a potential kind of like what you're talking about, you know, through the optimization. You go and find an effective one-particle operator that's self-consistent with that. And, you know, as chemists, we often say Hartree-Fock and self-consistent field interchangeably. You know, our chemistry source would call this--
52:22
So is it in fact the case that the Hartree-Fock is exactly-- that the state it's finding is exactly the ground state of something where you get rid of the two-electron interactions, just have electron in some arbitrary field, and then you find the field corresponding to the average positions of-- Well, like, yeah, like what he was saying, we don't quite get rid of the two-electron integrals, we just treat them in a mean field, in an averaged way, such
52:44
that you-- Isn't that the same, or are they the same thing? No? Okay. I don't know if the difference right here is just language or, but all of the and maybe we can follow up after, but, Yeah, It is folded in [MODERATOR] An effective averaged way.
53:11
Questions over there? [AUDIENCE MEMBER] You-- you start with the Hamiltonian that is that is uh, commonly used for chemical problems, che-- chemistry quantum chemistry, and then you reduce it to a qubit Hamiltonian at some point. If I see one of these qubit Hamiltonians, will I be able to guess that it came from
53:38
from one of these che-- chemistry-related Hamiltonians? [BRYAN OGORMAN] That's a very good question.
53:47
I think probably not.
53:50
I mean, at least not obviously. I mean, it has you know, the-- it's similar to all of these other situations where we reduce, you know, a Hamiltonian of one sort to a Hamiltonian of, of another sort. And, you know, we get some special subsets of the Hamiltonians in the, the target space
54:12
But it's maybe not so easy to-- to recognize that this is the case. Although it would be interesting to consider that itself as a computational problem and see if, you know, is it easy or is it hard? I don't know. So another [MODERATOR] Question [BRYAN OGORMAN] Here. Yeah. But let me just say one more thing about that, is like it's very important to remember that we did this transformation, and not to think of things in the qubit thing,
54:36
but to really think about things in the concise representation that they came from. You know, a lot is obscured when-- the-- Like, if, if I wanted to express the Hamiltonian to you, I would ex-- express it in the fermionic representation. I wouldn't, you know, translate it to a qubit operator, eh, and give it to you, and say, "Yeah, this is, you know, a Bravyi-Kitaev encoded Hamiltonian in this basis." You know, it would lose-- It would obscure a lot of
55:00
things that you could otherwise probably take advantage of. [AUDIENCE MEMBER] Sorry. I was just wondering-- --if Could you use Hartree-Fock to kind of set the ground state for some of the qubits, and then do the configuration interaction using phase estimation? Or, would it not, would there be no potential use case for wanting to combine the approaches? [BRYAN OGORMAN] Yeah. I think all of the approaches that I mentioned could be combined in arbitrary ways. And yeah.
55:31
Uh, so I'm not sure e-- exactly what you were proposing, but there, there are methods where, you know, we fix parts of the Hilbert space, and we you know, try to optimize over the other ones. And in fact, when we look at doing things on a quantum computer, oftentimes, we do some first approximation on a classical computer of a very large system, and then we notice that there's some central core that's strongly correlated, and it's only that central core that we represent on the quantum computer.
56:04
And the effect of the other electrons that we fixed is introduced in the classical way. Yeah.
56:13
[AUDIENCE MEMBER] Thanks. What is your thought about quantum embeddings and the promise for that approach?
56:19
[BRYAN OGORMAN] I don't know. Um, that, that's something that I've only recently been looking at, and, and Yeah. I haven't fully appreciated, you know, what the promises and challenges are there. But it's definitely an interesting direction. [AUDIENCE MEMBER] Yeah, I can try a different question, then. [MODERATOR] Okay. [AUDIENCE MEMBER] Sure, yeah. You presented things like Hartree-Fock in the second quantized picture. [BRYAN OGORMAN] Mm-hmm.
56:43
[AUDIENCE MEMBER] How often is the first quantization picture used in classical quantum chemistry simulations in comparison, or is second quantization the dominant approach? [BRYAN OGORMAN] So I, you know, will defer to someone who does on a daily basis, you know, classical computational chemistry. But I will say that while I focus on second quantization there are cases when applying quantum computers that the first quantized rep-- Or The first-- Yeah.
57:12
The first quantized representation is also useful. So there's this very nice example in using the plane wave basis, where if you look in first quantized representation, you know, you get this enormous saving in the number of qubits that you need, because you can represent the state in binary. And, you know, this is always the case, but now because it's the plane wave basis, you can actually compute on this basis, because the basis is indexed by these momentum states.
57:45
And so you can implement things like the kinetic energy not, Like, the index is not just some arbitrary label. It's literally, you know, in a binary representation, the momentum. And so you can actually compute the, um-- You can compute the, the parts of the Hamiltonian, as opposed to just looking them up, or using QRAM, or whatever like that. Right? And so there are definitely cases where it's very useful to look at first [AUDIENCE MEMBER] Quantized. Okay. But then you have to antisymmetrize.
58:14
[BRYAN OGORMAN] So in this particular case, you know, it's not so hard to set up the Hartree-Fock states in first quantization. You know, there are you know, nice and pretty simple circuits that'll do it for you. Um, and then as long as you-- Uh, so the, the Hamiltonian, you know, compute, commutes with the the swap operator. And so, you know, as long as we ensure that our initial state is antisymmetrized, and the only thing we do is, you know, evolve in time under parts of the Hamiltonian
58:49
that respect the symmetrization, then it stays symmetrized. And then maybe, you know, at the end, we can do some error correction. You know, we can see maybe this isn't quite as symmetrized as it should be and try to sort of snap back to the nearest antisymmetrized state.
59:08
[MODERATOR] Great. All right. We'll ask more questions over-- Uh, Daywan? [DAYWAN] Related to that, I mean, have people thought about imposing that symmetry through measurement? So, ignoring the fact that you have to symmetrize your operators? Evolving for a little bit, but then projecting into just the symmetrized subspace? [BRYAN OGORMAN] Yeah, yeah. So there's there's some work by
59:33
Babush, R-- uh McClean, and Rubin that look at, um-- So, I, I didn't have time to talk about this, there's this, like, end representability problem where you're given, like, the reduced density matrix-- matrices. And you're interested in, you know, are these consistent with each other? Is there this global state? And I think that those sort of questions are related to what you're asking about. But we can talk more offline. [MODERATOR] All right. I have to make sure people get their break, so.
1:00:02
Yeah. Thank you. Bye. (applause) (audience murmurs)
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