>> Jarrod McClean: Alright. I'll go ahead and get started and first let me quickly explain the
pictures I have here on the slide which are, is that the alternate title for this talk is How to Do
Quantum Computing on the Computer in Your Lab Rather Than on the Computer in Your
Dreams, so I hope to tell you a little bit about a practical project that we've been working on in
collaboration with a group in Bristol and also with Peter Love who is sitting here in the
audience, in order to use quantum resources to their maximum value that we happen to have
today in the laboratory. A little bit about myself, I'm Jarrod McLean. I'm a graduate student in
the Alan Aspuru-Guzik’s Group at Harvard. He would've liked to have been here but he just
recently celebrated the birth of his second child and is at home caring for his newborn along
with his wife Dori, so he's very excited about Microsoft's interest in quantum computing and
sent me as his liaison. Why quantum chemistry? Marcus was nice enough to elaborate on the
difficulties of this particular approach that I'm about to describe but really the promise of
quantum chemistry is that I start with some large molecule and I don't really have that much
other information about it. Maybe I know where its nuclei are and I'm interested in it, so what
theory can give me in practice a sort of understanding about these types of molecules? Maybe
I want to know something about its optical properties. Why does it absorb light in the visible or
UV or x-ray region? If it's a chemical reaction I want to know what intermediates are
important, how important are they. I want to know the relative energies, how things bind and
whatnot. I want to know how molecules interact with surfaces and so what can this lead me
to? Like any good theory, the hope is that understanding would then lead to control. So if I can
understand why certain things absorb light, I can start to predict new materials for say organic
photovoltaics. If I can identify key intermediates in certain reactions, maybe I can start to
inhibit the growth of unfavorable proteins or mis-folds in proteins that cause diseases like
Alzheimer's. And if I can understand both reaction coordinates and how things interact with
surfaces, I can start to get a grip on catalysis and maybe finally get platinum out of catalytic
converters. So the electronic structure problem, I think it was Paul Dirac who had this famous
quote which was that "the underlying physical laws necessary for the mathematical theory of a
large part of physics and the whole of chemistry are thus completely known and the only
difficulty is the exact application of these laws lead to equations much too complicated to be
soluble." And so of course he was talking about the time independent Schrodinger Equation or
in some instances a time-dependent one, but the promise is that I can take some complicated
physical object like this benzene molecule here with its many electrons and map it into a very
simple object over here which is just a mathematical operator. So in its most natural form of
Coulomb repulsion it looks a little bit daunting, but really all it is is kinetic energy, how fast
things are moving and potential energy how like and different charges repel or attract each
other. So in the call for slides asked for a little bit of background on how we traditionally solve
these problems and so the traditional approach is taken in quantum chemistry is at first you fix
the nuclei as classical and then rather than think about, if you think about all of these positions
of the electrons, you could think about gridding them and having densities in different places.
And very quickly for even small molecules this becomes entirely intractable, in order the
exponential space that you would need to store all of this, and so you can simplify this
dramatically. It won't remove the exponential scaling by using a little bit of chemical intuition
and say maybe a hydrogen molecule isn't so different from two hydrogen atoms sitting next to
each other, so you introduce the idea of atomic basis sets. And then the one last thing that that
Hamiltonian did not fully capture is that there is inherently a symmetry problem which is that
electrons are anti-symmetric and if you've ever done Diffusion Monte Carlo you know that you
can easily fall into the trap that the actual ground state of that last Hamiltonian is bosonic and
so you can sort of capture both of these problems at once by moving to this form that we've
seen so many times already today which is that I quantize things in terms of spin orbitals,
introduce an atomic basis set and I can move to the second quantize representation that also
encodes my anti-symmetry, which is a very nice trick. But the problem still scales exponentially.
So the next approach is usually we say okay, well if we take some target electron, what can we
do with that? Well, in the mathematical sense we say maybe it's given by a single
configuration. It turns out that this is equivalent to the physical idea that I smear out the
density of the rest of the electrons and put a single electron in this. And this reformulates my
problem as a mean field problem which is efficient to diagonalize and you get out an answer
which is just the vacuum state populated in your new set of molecular orbitals which
diagonalize this particular term and chemists like to diagram this because they don't like math
so much as these simple pictures over here where each spin orbital is just a line and I choose to
occupy it or not occupy it by an electron given by these arrows here. Right? But what's the
problem with the mean field? I'm going to focus on the mathematical idea which is that I start
with just this single configuration here, but there was no reason that I needed to truncate my
space in any extent and so actually all of these other configurations are allowed too. I have a
single vector in this anti-symmetric space, when I could have had all of these. So conceptually
what is the easiest way to include all of them, computationally not the easiest, but that's called
configuration interaction which is that I re-express my Hamiltonian in terms of all of these many
configurations and I diagonalize it in this linear basis and this is sometimes called exact within a
basis and it's more commonly known as full configuration interaction and unfortunately it
scales exponentially in the size of the system, the space does. So here’s some alternatives and
some of the scalings that we have here and you can see that by 2015 this full CI is not doing so
well. This is from an old paper so there's some additions that you have to make to it, DFT is
doing reasonably well in terms of scaling but we know that it has many qualitative and
quantitative errors. And of course you have to add tensor networks somewhere in here and I
don't promise that that line is exact because it's an artist’s rendition, but you can see that we're
making progress. Many of these methods are approximate that scale well, but what we really
like if possible is to do this system exactly, so what can we say about that? Just in case those,
that sort of wasn't to your interest, what can I say about the summary of why quantum
chemistry, is that it tells you almost everything about a quantum chemical system if you can
solve it and if you can solve these large eigenvalue problems exact chemistry on the scale of
catalysis might be within our grasp. So where does quantum computing come in? It comes in
by an algorithm introduced, well, the one that we've already seen today. Quantum phase
estimation and I think it was the first my boss Alan Aspuru-Guzik who decided to apply that
particular algorithm to chemistry and see that we can achieve some of these things, some
instances in polynomial time. Just to give you a quick cross-section of our group's research, we
work in many areas and in particular we do fundamental research in both electronic structure
and quantum computing and their interface, and to characterize the quantum advantage that
was originally sort of sought out, you can see that this problem full CI, the classical costs due to
its size is exponential and if you assume that you can efficiently prepare the initial state, this
evolution cost is somewhere on the order of n to the fifth, but maybe after Peter Love’s talk will
have to amend this slide, so we will see, to an order n to the fourth or n to the four log n,
perhaps, so we will see. And this is just a cross-section of some of the continuing work we do at
the interface of quantum chemistry and quantum computing including some of our
collaborations with experiments that allow us to literally simulate some chemical molecules on
other chemical molecules which is sort of exciting if you think about it. So we work with
photonic groups, both Philip Walter’s group and the work I'll talk about later is with Jeremy
O'Brien’s group and this is John Feng Du’s group in China. So a tale of two algebras, a little bit
more introduction, so I talked, have been talking so far about fermions and thus far today we
haven't seen the Jordan-Wigner Transform so I didn't exclude this slide and on a quantum
computer we have qubits. On the left something that is distinguishable and on the right
something indistinguishable and anti-symmetric. So it turns out that they are isomorphic by the
Jordan Wigner Transformation. That is that I can go ahead and just write each of these creation
operators that populate a state as a simple spin flip on some sort of spin reference state. You'll
notice all of these Sigma Zs that tend to accumulate are only there to keep track of the plus and
minus sign due to anti-symmetry. What I want you to notice about this is what's relevant for
this talk is that it preserves the number of terms in a Hamiltonian but in general not its locality.
What I mean is that on each one where before we had terms that acted on it most for spin
orbitals, as a result of these large strings of Sigma Zs we might have almost totally non-local
qubit operators and these are the origins of these Jordan Wigner ladders that people talk about
so much. Quantum phase estimation we've got a nice detailed introduction by Krista earlier. I
won't be going into it in any depth except to cartoon it as the following three phases.
Preparation, I'm not actually going to assume that I'm preparing an eigenstate; I'm going to
assume in this talk that I'm preparing some state which I hope is the best approximation of my
eigenstate. I'll evolve it using all of these series of unitary operators and finally measure it. And
this is the approach that's been taken almost universally in the work on ground state quantum
chemistry with the exception of some alterations by adiabatic state preparations and so if you
are familiar with the bounds these are a little bit outdated and there is a graphical version of
the actual numbers on the next slide, so don't worry. But what I want to call your attention to
and then I'll talk about later is that the dominating factor in the success of your measurement
actually depends on this guess state that you made down here and that's going to be pretty
relevant for what we work on. So our group has studied what the actual qubits and qubit
operations will end up being, so these numbers are fairly large for say a molecule like benzene
or otherwise in a sizable basis. You'll see that the relevant number here that I am talking about
is the number of two qubit operations required for Trotter decompositions to chemical
accuracy. And you will see that they grow dramatically. We have numbers on the left like ten
to the 11 or ten to 13 and we saw some nice time estimates earlier on what that might mean in
terms of months or even years that you need to keep your fragile qubits in complete coherence
in order to gain your quantum advantage. So this is probably one of the primary technological
challenges is that if you wanted to implement the quantum phase estimation algorithm you
would need to keep your qubits coherent and accurately perform those gates for on the order
of months or days or even years right now. So we've also quantified the qubit requirements
here where you see that in different basis sets you can scale save maybe cholesterol that you're
interested in and you may want to do it in a compact triple zeta basis. Maybe you need 1500
basis functions and in a compact representation that might need 900 qubits. We've also
thought a little bit about error correction. I haven't talked about it. I won't talk about it too
much after this in our scheme. We quantified the fact that the current standard, Solovay-Kitaev
could be bested in terms of a little bit of resources with just the phase kickback algorithm which
is analogous to quantum phase estimation and the tricks that you use there, and that if you
include error correction these requirements can increase drastically. And so I'm sure that a
better coverage of error correction will be given by Ken Brown later in this series, but for now
we're not going to concern ourselves with it. Yes, question in the back.
>>: [inaudible]
>> Jarrod McClean: Oh sure, this particular paper compared several different methods for
constructing circuits out of fault tolerant gates, and so if you talk about I want to do -- actually
this is the lithium hydride molecule and so if I quantize that in a minimal basis and then perform
the Trotter expansion of the time evolution, then if I want chemical accuracy using these
different methods of constructing a fault tolerant sequence of gates and the primary driver for
fault here is these controlled Z operations. And if I use a Solovay-Kitaev scheme I can use a
fairly small number of bits but a very large number of operations. But a new trick that was
introduced in this paper was actually performing these controlled rotations via a phase kickback
scheme and we compared to other competing methods like Fowler and the equivalent
representation in the first quantize, and the Fowler one looks a lot better but there is a secret
third axis here which is the exponential classical pre-computation required in that scheme.
Yes?
>>: Is this [inaudible] Trotter further full-time, for the full [inaudible] estimation? This was
1000 qubits?
>> Jarrod McClean: Uh-huh. So the circuit depth is actually the full evolution that you would
need for one pass which comes up to about ten to the eleventh in some of these. Yes?
>>: [inaudible] for what time you [inaudible] it?
>> Jarrod McClean: We evolve it to the time required for the chemical accuracy precision in the
evolution, so whatever the corresponding time that you back out required to get the resolution
in the phase that you would want for chemical accuracy.
>>: [inaudible]
>> Jarrod McClean: Yes.
>>: How do you get only ten to the seven at gate depth when you have ten to the three qubits?
>> Jarrod McClean: Uh-huh, so that's sort of the trade-off in some of these algorithms is that in
particular this second quantize par, you can do tons of pre-computation in the end silicates and
this lets you trade it off a lot of computational complexity in the construction of the faulttolerant gates.
>>: Ten to the three qubits so ten to the three, so that's qubits [inaudible] that is -- so that is
the size of your final compute. It's not the number of orbitals that you have.
>> Jarrod McClean: Yeah, it's the size of the quantum computer.
>>: Oh, okay.
>> Jarrod McClean: So the orbital actually -- this is a very small system that, the number of
qubits used to represent it is on the order of 4 to 8 and so this is the additional resources we
borrowed to do error correction.
>>: Thank you.
>>: I just wanted to make a general comment, while this slide is up because I think it highlights
that there is a regime which is kind of the opposite of the one that the PS focused on in this
introduction is that it's quite possible that it's more realistic to think about what could be
usefully done with a million qubits if you're only allowed a total of a billion operations. You
know, because you might be able to build a quantum computer with a very large number of
qubits but the coherence times are such that you can only run it for a minute like that. So to
some extent these resources are fungible but not completely, so there might be completely
different ideas that could be implemented kind of in the lower right corner of your picture
[inaudible] the left.
>> Jarrod McClean: I think that's an exciting option and actually these skewed lines I didn't talk
about was that some of these algorithms even have adjustable parameters such that you can
decide can my quantum computer cohere things for longer or do I have a lot of extra qubits
around. So I'm going to talk about now an entirely new perspective on the problem which is to
identify that currently you have some task that you want to perform, say eigenvalue estimation
phase estimation. And you then go and design an abstract circuit for it. You get together with
your computer science friends, prove that it's optimal; everyone is happy and then enters this
question mark on whether you can put it in your quantum blue jean Q over here and get out
the answer to life 42. So what's the problem with this approach right now? The problem is that
the general or optimal solution can require millions of gates as we were talking about before.
It's very hard to stay coherent for that long. So we are going to propose a slightly alternative
solution which is I'm going to consider the task and the architecture at the same time and I'm
going to use it to create a very simple prescription for getting an approximate answer, and this
will be the approach that we take for the rest of this computation. In order to maximize
quantum resources you want to ask what is an easy task for a quantum computer? Just about
the easiest one you can have is a measurement in the Sigma Z basis, so later when we show a
photon implementation of this I will just ask if this came out of slot one or slot two. Tandem to
this you can then do correlated measurements by looking at where do they come out in
concert. So it's efficient to perform on any prepared quantum state. It's about the easiest
thing you can do, and in general it may be very hard to compute this expectation value for a
classical representation. How are we going to formulate this into an actual algorithm? You go
back to the very basic day one of quantum mechanics and you construct a variational
formulation that my ground state eigenvector that tells me all of my interesting chemical
information is the thing which minimizes the expectation of this Hamiltonian. So I can write any
Hermitian Hamiltonian as it turns out and we have an exact mapping for the quantum chemical
one as this decomposition over tensor products of poly matrices, where these Hs are just these
simple classical numbers that we know before performing the computation and these poly
operators are the ones you're probably used to. And then by the linearity of quantum
observables we see that its expectation can be written as this classical sum of all of these
expectations that I can perform very easily on any prepared quantum state. So we take what's
easy for a quantum computer, so computing these expectation values and things that are easy
for a classical computer. Not in the sense that they are hard for a quantum computer, but in
the sense that my resources are essentially infinite on a classical computer right now in
comparison to my quantum resources, so I put on the classical computer what I can. This
suggests a hybrid scheme. If I can parameterize by quantum state with classical experimental
parameters, compute all of my averages on the quantum computer and update the state with a
classical algorithm, then I can sort of utilize the best of both worlds that I have available to me
right now. So the computational algorithm diagram can be seen as follows. I prepare some
quantum state and I'm going to get more into what defines that quantum state in a slide or
two. I send it into any number of quantum modules that I use to then compute my averages,
sum them up on a classical computer and then decide a new set of parameters for my next
classical state. For example, these might be angles on a beam splitter or things like that that
you would put in your lab. So from this idea you sort of think about what is the state that I'm
preparing and you have to sort of think about maybe a fundamentally new idea if you want to
do this in the most general sense which is a quantum hardware state on ansatz. So normally if
you talk about ansatz in modern chemistry you have something like a coupled cluster or maybe
a matrix product ansatz that you can write down and characterize, compute observables on and
everything is okay. On a quantum device which is inherently complicated, you maybe have a
set of knobs. You can think of them as analog or maybe it's the duration I apply a pulse
sequence for. Maybe it's the strength of my magnetic field, but the point is you have some set
of experimental parameters which will compute a quantum state which may not be easy to
represent classically. So the advantage is that if you have built a really complex and intricate
quantum device, you can use that complexity to your advantage to represent states that might
be very hard to write down classically. It will always satisfy a variational principle due to the
fact that you perform unitary evolution in ideal case, perhaps not if the machine falls apart, but
coherence time requirements are set by the device not the algorithm. I know how many
operations my say my ion trap or my photon apparatus can perform before it decoheres. So I
just say that's the state I'm going to prepare and I'll never have to exceed that requirement just
because I'm trying to match the confines of some optimal algorithm. So just to provide a
corollary to this, this is actually the result that Mike Friedman was talking about earlier, an
exciting result by Aaronson and Arkhipov proved that under reasonable assumptions about the
complexity hierarchy, even approximate determination of the probability results from a simple
linear optical network whose construction is known is not efficient for a classical computer. So
what optimistic consequence can you draw from this? Is that with only a few tens of photonic
qubits the quantum averages sampled on an architecture like this might not be efficiently
sampled by a classical computer. That's an exciting prospect for the new states you can think
about manipulating that are not so easy to write down on a classical computer. In case you are
scoffing because all of your methods get excited state energies. I'm interested in absorption
spectra. There is a trick we can use to fix this variational problem. It's pretty old and actually
was introduced by a quantum chemist I believe. It's obvious when you see it which is just the
folded spectrum method, so before the algorithm was getting us some eigenvalues. So if these
are just the eigenvalues present in a quantum system characterized by this energy axis. Before
I was talking about the ground state, but you can introduce some shift parameter which will be
this blue dotted line and as long as it's closer to my target state which is now red, the effect of
this shift is to fold the spectrum up and make the new ground state my target state. So this
trick has been used for some time in quantum chemistry for interior eigenvalues. There’s other
more sophisticated tricks, but this is about the simplest one you can write down and it's very
easy to put on a quantum computer. It isn't without perfection. It roughly squares the number
of terms in your Hamiltonian so that it will have polynomial effort and it can accidentally cause
degeneracy in the ground state if you accidentally select a shift which is directly in the middle of
two eigenvalues, so full disclosure. So wouldn't be exciting if we just postulated it. The reason
we focused on doing so much work with minimal quantum resources is that we like to work
with experimentalists and right now they have on the order of somewhere between one and 10
qubits depending on who you talk to, sometimes more sometimes less and so the model
system we did in chemistry was originally we worked with H2 plus, or H2 which due to
symmetry was a 2 x 2 matrix and now we are doing helium hydride which is excitingly a 4, 4 by
4, and so we have a compact representation for this particular Hamiltonian and we built it
excitingly. So I'll give you a tech close-up on the next slide. So what we have here is actually an
integrated quantum photonics device that you put entangled photons into and to give you a
sense of size, these are single pin connectors here coming in from your spontaneous parametric
down conversion source. These are about the size of the end of the pencil, so not a beefy end
including the light brown, so these are fairly small and it's an exciting step towards
miniaturization. These are made by a company for the O'Brien group in Bristol and it's a nice
transition from the laser tables that you traditionally see in a lot of labs that you can really start
to get a mental image of how these might fit into or on top of your computer. We called it a
QPU following the tradition of CPU to GPU and now QPU that perhaps your quantum device
isn't fantastic at everything, but there are certain tasks which it is exceptional at that you can
export these operations to that you'd like to perform. To elaborate on the physical description,
each, the state is characterized by a series of beam splitters whose parameters you change
actually by resistive heating of the silica, and this changes slightly the length of the beam and
this allows you to perform different quantum operations. And the features of this particular
chip which they had fully characterized before we worked with them is that it has
reconfigurable one qubit gates and it can produce an arbitrary two qubit pure state with fidelity
better than 99%, so that's a great statistic if you have a state. And to think about the quantum
state, as I was mentioning before in this particular case it's determined by this fixed sequence
of parameterizable optics and the consequence that you deal with on all of these is that if the
apparatus is hard to simulate classically which makes it useful, then it's hard to know which
states it can efficiently sample from. And you can think about that from a negative or a positive
standpoint. I like to think of it from the positive standpoint in the sense that it's very exciting to
work with new objects that are very hard to write down, and you can even think about the
black box that I drew on purpose as a black box a few slides go and taking say a machine
learning type approach that couples the best of your knowledge of classical minimization to this
quantum hardware to produce genuinely new states or a further extension like Mike Friedman
was talking about was that if a quantum optimizer is the best you can do, maybe I optimize my
quantum state with a quantum optimizer. It's just a thought. So what did we get in
experiment? Experiment is messy, so we measured the lowest round state eigenvalue for a
series of separations of this molecule and we computed the energy at each of a given set of
parameters here. You can see that in many cases we are within chemical accuracy but in many
other cases we are not, and the result of the origin of a lot of this noise is simply that you have
a lot of photon detection there. You have to take long statistical averages and the fact that we
run a minimization on an inherently noisy surface that's not only noisy but it drifts actually, and
so this is a sample optimization at one geometry. This is not a direct minimization in the sense
that we don't use a gradient descent, so it jumps up and down because it is a simplex
optimization. You don't want to have to deal with the fact that when I have a set of parameters
and return to the same space that it might not be exactly the same result. But the exciting part
that I want to draw your attention to is the colors of the dots which quantify the tangle which is
the concurrent squared and it is simply a measure of is my state classical or is my state
quantum. And you see that our algorithm takes advantage of the truly quantum states as it
progresses through the space to find the ground state which in this case happens to be very
nearly classical. Now I'm going to talk…
>>: [inaudible] energy in each of those steps?
>> Jarrod McClean: So the energy in each of those steps is given exactly by the prescription
that I had before, so I prepare my state over and over and over again. I perform the
measurements which correspond to this Hamiltonian here and I have the Gs that I've
precomputed which are classical, and each of these expectation values on my quantum state,
and then after I compute each one of them through many repetitions I average this result
together and that gives me the energy that you have. Here the red lines if that's what you are
referring to are theoretically known if you just diagonalize exactly.
>>: [inaudible] Hamiltonian times square root [inaudible].
>> Jarrod McClean: Yeah, yeah due to the averaging. Now I'm going to talk about that
reference that I made before which was the troubling part of the algorithm in quantum phase
estimation that a lot of people gloss over which is this inherent measurement problem at the
end of it that maybe I don't know how to prepare an eigenstate of an interesting system
arbitrarily. So this catastrophe was actually -- I'm not sure if it was named it, but it was
certainly referenced as such by Walter Cone in his Nobel speech for the reason you might want
to do density functional theory and I don't know that modern density functional theory gets
away from it, but the statement of the problem is as follows. So this is a toy representation of a
water molecule. Say I have, I know the exact state and I have some guess for the trial state.
Maybe I've done a variational calculation. Maybe I've done Hartree-Fock and say my guess was
pretty good and the overlap between the two which you remember I said the probability of
success for these measurements was related to the overlap, so maybe it's 90%. That's pretty
good, right? So now I think about two molecules and the tensor product of those together and
then I say okay, I just want them to not interact. I'm going to sit them next to each other, take
their composite weight function and try again versus the exact answer, and now you see that
the overlap is .81. So that’s not too bad, right? But interesting chemical systems have many
water molecules and if I take perhaps a very modest estimate of a hundred, then you can see
that I'm not really doing so well after all. I won't say that this is a terrible catastrophe in the
sense of normal quantum chemistry. Actually, strange it was named the Van Vleck catastrophe.
In fact he was the one who responded to this criticism and showed that despite this diminishing
wave function quality, observables in fact were only errant in polynomial in the size of the
system, but it is an interesting problem to consider if the main source of cost is going to be the
overlap. So I won't say also that this is a damning problem for quantum computation in general
because we've done some estimates and Sebre Kise [phonetic] has done estimates on how you
can improve trial wave function quality and overlap and the system sizes for which this starts to
become a problem in a quantum computer are far beyond the systems for which the qubit
requirements are currently a problem. But it is something certainly to address and we hope
that this approach that we've presented which is very cheap and adjustable to a quantum
system, can actually help to bolster that success. So before the typical approach that you used
in quantum phase estimation was the state you input was just the Hartree-Fock state. And so
now we're saying that you could bolster this guess for any future implementation of this by
simply implementing this variationally optimal quantum hardware guess. And the hope is that
by increasing that overlap you can actually increase the success of quantum phase estimation in
the long run also. So just to summarize, just by considering the problem and the architecture
we hope that it offers a new way to utilize quantum resources that are available today and not
in 20 years. We built a small-scale implementation and we have hope that maybe with a few
tens of qubits we can start to manipulate quantum states that would be otherwise hard to think
about classically, and we propose that it's complementary to phase information rather than
competitive. With that I'd like to thank everyone involved in the work in this project so, Alan,
myself and Man-Hong Yung from our group. This is the Bristol group which builds the
apparatus, Jeremy O'Brien over here, Peter Love who is featured in the second row and on this
slide and with that I'll take any questions.
>> Michael Freedman: Thank you. [applause]
>>: So Jarrod, in your -- since you are taking these operator averages, I guess you end up with a
double hit with precision relative to [inaudible]?
>> Jarrod McClean: So we take a hit with respect to phase estimation in the sense that we need
to perform averages likely for much longer. Are you talking about if I were to use it to bolster
phase estimation or on its own?
>>: So Mateus’s talk, right, so you end up with n to the 4 over epsilon pre-factor?
>> Jarrod McClean: Yeah, Uh-huh.
>>: So now you say you get n to the 4 over is it epsilon squared or…
>> Jarrod McClean: Are you talking about in terms of if I use this before quantum phase
estimation or on its own?
>>: On its own.
>> Jarrod McClean: On its own? So…
>>: Imagining actually that you don't even have to do the variational part. That you somehow
can give it a nice state.
>> Jarrod McClean: Okay, yes, if you somehow you have been given a nice state, so it actually
depends largely on the variance of the operators that you would then measure it on and so I
would…
>>: You know from…
>> Jarrod McClean: Yeah.
>>: From like people looking in [inaudible] studies or [inaudible] the backup space like if you
look at the electron sort of occupancy of 1% and then if you have these [inaudible].
>> Jarrod McClean: So I'm not just talking -- so not just in the sense of the theoretical sense of
the variance, but in the quantum sense and how accurate the technology is, so our major
limiting factor is not the theoretical sampling, but was rather the technological limitations of
the single photon sources which were frequently errant and the drift which results from
adjusting precise values via heating and so…
>>: You going to get Monte Carlo sampling errors.
>> Jarrod McClean: Yeah.
>>: I mean [inaudible]
>>: So that's epsilon squared [inaudible] right? Because to get epsilon you have to [inaudible]
[multiple speakers]
>> Jarrod McClean: So we are still Monte Carlo limited in that sense, yeah, but I was just trying
to get at the variances largely from the experiments.
>>: Sure.
>>: So you explained this orthogonality catastrophe but what was the solution to it? That went
by rather quickly.
>> Jarrod McClean: So there is no perfect solution, but our hope is that this quantum state
hardware ansatz, so instead of just taking the mean field guess, now I input the mean field
guess as an initial state and I use my quantum device to find a variationally optimal set of
parameters on my, say, ion trap and hopefully by improving the state of the guests then I repair
some of the overlap and you'll say that's a pretty bold assumption, but it's also -- so it's sort of
the assumption that's implied in a lot of the variational methods that are used in quantum
chemistry today which is to hope that as you decrease the energy you provide a better
description and greater overlap with the ground state.
>>: [inaudible] play that game and I ask myself a question and I can write down a classical
variational guess just naïvely as sort of a vector [inaudible] or something or assume [inaudible]
or something. Do you have sort of a rough order of magnitude of how many dates of
coherence I actually need to pull this off before I’m out of guesses that I could perform
classically?
>> Jarrod McClean: So in terms of gates of coherence I don't know an order of magnitude guess
for that. The qubit estimates that have been given by Aronson and Arkhipov for the
determination of permanence was I think in the order of high 30s or low 40s and you only
needed as far as I remember not that many linear optics operations, nothing even terribly
complex to produce a state which was…
>>: [inaudible] 30s is such a big number.
>> Jarrod McClean: Two to the 30 is not. Two to the 40, two to the 50 is considerable.
>>: Okay. [laughter]
>> Jarrod McClean: There is that gray area right there. It also hits on one of the largest open
questions that I hear everyone ask which is what is the threshold for which when I have that
number of qubits I can no longer verify my answer classically and what do I do in that case. You
can't compute it classically but your quantum computer is giving you a number. What can you
do about it?
>>: Governments? A
>> Jarrod McClean: Yeah.
[laughter] [multiple speakers]
>>: You look at the system you have two electrodes.
>> Jarrod McClean: Yes, two electrodes.
>>: [inaudible] between this one and H2 [inaudible] experiment changing the [inaudible]
universal circuit for two electron [inaudible] system. I can go to H2 or I can choose for example
hydrogen [inaudible].
>> Jarrod McClean: Uh-huh. So this can -- so what was your question, sorry?
>>: The only difference between this one and H2 is the [inaudible].
>> Jarrod McClean: Yes.
>>: What is the [inaudible]?
>> Jarrod McClean: So the nuclear charge only…
>>: I can build any two electrodes [inaudible] system and instead of just going one by one.
>> Jarrod McClean: So the nuclear charge only comes in in the sense of the molecular orbital
pre-factors that we compute, so I don't know that you could construct such a trivial
extrapolation for any 2 x 2 system, but this particular one, and you could do things like LAH
which we also considered in a slightly less compact way. The reason we chose HEH plus was
because it was just that it was directly within the qubit confinements which were at two qubits
for this particular photonic system, but I don't know that you could write down in a naïve way
that the molecular orbital was just a function of the charge between the two. Maybe it's
possible.
>>: So you are saying to go from this [inaudible] H2 is this getting the charge?
>> Jarrod McClean: Yeah.
>>: You know, between the two?
>> Jarrod McClean: Uh-huh.
>>: So you don't have a new system. It's like the same but you just run this one [inaudible].
>> Jarrod McClean: Yeah, yeah so you could absolutely write down H2 in exactly this same
compact basis I did here which is that these Gs would scale as a function of the charge if you
could continuously deform the charge. I don't know if that's what you're talking about, but you
could certainly write it that way and the way it was originally formulated when Alan did H2 with
some of the original experiments. The difference here is that we have slightly less symmetry in
HEH plus then H2 and as a result you can no longer have these 2 x 2 blocks that form a block
diagonal Hamiltonian so the extra complexity comes from the lack of symmetry there. Does
that answer you?
>>: [inaudible] some energy [inaudible] with respect to your variational parameters. My naïve
guess is that would be an extremely complicated [inaudible] over.
>> Jarrod McClean: Yeah, so we don't make any claims that this solves the QMA complexity
problem of the ground state. So in that sense…
>>: [inaudible] much simpler question. For a simple molecule because of the way you are
parameterizing your variational landscape, do you have any sense for whether one could
reasonably sort of find [inaudible]?
>> Jarrod McClean: So to give you a sense of optimism, the minimization that we did here
started from a completely random state, so we used no initial knowledge which is why they
tend to oscillate so wildly. If we had started from the Hartree-Fock in this particular one we
would've already started with 98% overlap with the ground state. And so we were able to
reasonably efficiently find it for this system. Whether that is a prescription that can be carried
on to very, very large systems, of course, as everyone here knows nonlinear optimization is
quite hard, but it's also a problem that's been used quite successfully in quantum chemistry
namely in Hartree-Fock, MCSCF, coupled cluster theory and so there is some hope.
>>: [inaudible]
>> Jarrod McClean: Huh?
>>: There the variational parameters are physically motivated in a way that this isn't, right?
This worries me a little bit.
>> Jarrod McClean: Yeah, yeah, but I meant you could imagine constructing analog devices and
that is sort of some of the beauty of quantum mechanical simulators is that in an ion trap I
actually have the two ions next to each other so to some extent I can vary these parameters in
a very physical way. Say the distance between maybe two ions and I want to control it with my
trap string; that becomes a very real parameter, almost more physical than like a cluster
operator.
>>: How many parameters do you have in this?
>> Jarrod McClean: To produce an arbitrary two qubit state this particular photon device needs
six parameters and so these are just the couplings that you have between the different
waveguides, so these first six are used for the preparation and these last ones are used to do
rotations to prepare the measurements in the basis that you want. Yep?
>>: It seems like you could, I mean not with this [inaudible], you could make the device that
tries to work out chemical intuition, right?
>> Jarrod McClean: Yeah, and actually we think that's a very exciting direction to go in is the
design of devices that could very easily, or very naturally represent problems that we are
interested in, and so while we used this device because it was already fully characterized in the
sense that when you receive one of these quantum integrated circuits you can't just start using
it. You have to verify that each coupling is exactly what the manufacturer said. I produced my
fidelity with exactly the precision that I expect, and so that was the reason for using this device,
but we think it's actually a much more exciting direction to design new devices with this
thought in mind that you could have physical parameters.
>> Michael Feedman:
>>: Thank you again. [applause].
>> Bryan Clark: Okay. So I'm going to talk about Quantum Monte Carlo and sort of the
approach to electronic structure. I am a Quantum Monte Carlo person primarily, also some
other numerical methods, and so I'm going to tell you mainly about this, but I have three goals
really for -- my microphone is still working. Yes, okay, three goals for this talk. So goal number
one is just to give you a sense for the classical algorithms, so give you an idea for how these
Quantum Monte Carlo algorithms work and what we can do with them. Goal two is to say the
following, suppose we want to leverage our classical algorithms and sort of promote them to
working on a quantum computer. Can we take advantage of some of the tools of the quantum
computer to make our classical algorithms more efficient? So you might ask why talk about
this. Well there are two reasons. One, as Matea said, some of the current ideas for doing
quantum simulations on a quantum computer are slow and so maybe if we can kind of build a
toolbox of other ideas then maybe some of those pieces of that toolbox will be faster. The
other reason to sort of talk about this is we have a lot of algorithms for doing this and they
appear to be polynomial, but it's not obvious actually that any of them are clearly polynomial. I
mean they tend to rely on things like you have some quantum Markoff chain and it mixes fastly,
or you have some adiabatic path but it never hits a first order phase transition, or as Jarrod was
talking about, you can write down a variational wave function and it has high overlap. So it's
not even clear we have polynomial algorithms much less fast algorithms. So I'm not going to
answer anything really in this goal. I'm really going to sort of raise questions. I'm going to say
look, here are some questions that we might ask, can we do something like this on a quantum
computer. So this is going to make my talk a little schizophrenic because I'm going to be talking
about classical algorithms for quantum simulations and then ideas on how we might begin to
think about quantum algorithms for quantum simulations. So to avoid some of the
schizophrenia, whenever I'm talking about quantum computers simulating quantum
simulations I will notate it with this QC here. The goal here is to avoid any confusion about
when I'm talking about classical algorithms or quantum algorithms. And my final goal is to say
look, we are talking about doing real or quantum simulations on quantum computers so we
should ask what is the state-of-the-art or the kind of practical applications that one can do on a
classical computer, so that will tell us sort of where the niche is for quantum computers. So
these are my three goals today. So primarily I'm going to be talking about electronic structure
so we are sort of at the starting point of electronic structure. In electronic structure you really
write down the cool Hamiltonian. We saw this earlier, some kinetic energy, some coolant term
and then typically you think I have some set of six nuclei and they produce some potential. And
there are lots of games you can play starting with this Hamiltonian. The game we will play
today is very simple. The game we will play is we say we want to compute properties of the
ground state of this thing. So that's sort of the focus of what I'm going to talk about. And then
you can ask classically someone hands you this problem and says what are you going to do and
Jarrod actually had a very nice sort of slide where he said okay, here are how different things
scale and you look and you say look I want to do a big system. What am I going to do? And the
answer was sort of you do DFT. The DFT isn't very accurate and the next thing on the slide is
you do Quantum Monte Carlo and that's mainly what I'm going to talk about today. I'm going
to talk about doing Quantum Monte Carlo. You can do other things. You can do perturbation
theory, DFT, DMFT but I'll talk about Quantum Monte Carlo. At least those of us in the
Quantum Monte Carlo community like to think of Quantum Monte Carlo as sort of the de facto
standard for highly accurate albeit approximate calculations for systems that are medium to
large size. And I would say that is the general consensus. You can think of Quantum Monte
Carlo in a lot of ways like a better DFT. It scales much like DFT. It parallelizes really well. It's
still sort of order of magnitude slower than DFT, but it still is sort of on the same framework as
DFT. So that's what we will talk about today. And we will really talk about doing Quantum
Monte Carlo in two different ways. So in the one case we say look, we are interested in doing
an exact Quantum Monte Carlo. This is going to cost us an exponential amount of work, but for
a small system we can do it. And then will say look, we would actually like to do big systems so
we’re going to do something approximate. We are going to guess a wave function and see how
far that can push us. Then I will talk about sort of a combination of these called fixed node
Diffusion Monte Carlo where you do a little bit of both. So that's the framework for today's talk
and in the interim I'll be talking about applications and ideas for how to promote some of these
things onto a quantum computer. So people should ask questions while I'm talking, so don't
hesitate to do that. So what's the general framework for how Quantum Monte Carlo works?
The first thing you say is you say the following. Look we want the ground state. We want sign
not, so how do I get myself to the ground state? Well, I start somewhere, kind of some place
arbitrary and I multiply by e to the minus beta h for large beta. For a finite system, for a large
enough beta doing this matrix multiplication drives me to the ground state. So you say look,
this is easy. This is just matrix multiplication. Why is there all of this trouble? And really there
are sort of classically two problems with just very naïvely writing down a matrix, multiplying by
some arbitrary vector and getting the ground state that we want to work with. And those two
problems are the following. One is that the matrix elements of this matrix, e to the minus beta
h for large beta are hard to evaluate. They are slow to evaluate. If you say look, I want to know
what Re to the minus beta h R prime is it would take a long time to figure out. And the second
thing is that these matrices are large. They're essentially exponential in the number of
electrons and so they are difficult to work with. I mean it's difficult to multiply exponential
things. So it turns out that stochasticity, randomness can actually help you with sort of at least
this problem, when your matrix is too large to multiply. So if you just gave me a black box, even
a quantum computing black box that quickly evaluated matrix elements from e to the minus
beta h, then I can add to that some randomness and compute any property of the ground state
that I want. I mean it tells you how to do that here. You add the identity. You sample from the
states with some probability and you measure the observable that you care about. So okay,
classically we know how to fix one problem. This matrix is too large so instead of actually
multiplying it, we will do something random. We don't know how to solve this other problem.
The matrix elements are too slow to evaluate. You might ask the generic question can a
quantum computer do this. Probably not. No one knows how to do this, although there is a
sort of work by people whose names I can't pronounce that says look if you want the diagonal
entries of powers of sparse symmetric matrices, in essence Hamiltonians, that's something that
you can do reasonably fast. So maybe we can leverage this idea to build something that
doesn't exponential fast or maybe we can Teller expand. Maybe there are games we can play.
So this is sort of the kind of thing that we might want to think about to say look, classically if we
can solve this, if we had a black box that did this we would be in good shape. Can we imagine
building such a black box? But okay, let's step back into the classical world and ask we can't
solve problem number one, so what are we going to do? Let's start over again. Let's say okay,
that didn't work. We solved one problem but not both of them, so what are we going to do.
Well what we're going to do is we are going to take this matrix and we are going to split it up
into a bunch of matrices, so instead of e to the minus beta h for large beta, we will do e to the
minus tau h, times e to the minus tau h, times e to the minus tau h over and over again. So you
might say okay, what did this do for you? And it did the following. This matrix is hard to
evaluate matrix elements from quickly, but this matrix it's easy to evaluate matrix elements
from quickly at least to very high accuracy, so there's two easy ways to see this. If you are a
physicist, you say look, this matrix is large beta. Beta’s inverse temperature, so it's a low
temperature thing. Low temperature thing, quantum mechanical, quantum mechanics is a hard
thing to deal with and we don't know how to work with that. But here we've split up beta into
a bunch of little pieces. This is small tau that's high-temperature. Now high-temperature thing
is very classical. I mean we are masters at classical physics, so it shouldn't surprise us that we
should be able to write down a very good approximation of this classical object. From a
mathematical point of view this statement is also very simple. This Hamiltonian is the sum of
some operators that don't commute, kinetic energy plus potential energy, but if tau here is
small the error that you are making is order tau squared and their commutator. So okay, and
the limit is a very small tau and we can write down good approximations with this. Okay, that is
the mathematical statement. Okay. So that's nice. That solved problem one when the matrix
elements are too slow to evaluate, but we still have problem two, the matrix is too large to
multiply, so let's ask two quick questions. Can we use quantum mechanics to solve that
problem and before we found out that we could use classical stochasticity to solve that
problem, can we still use classical stochasticity to solve that problem. So these are the two
questions that we want to ask ourselves. So let's start with quantum mechanics. So you might
ask look, this is essentially a sparse matrix. It's approximately 1 minus tau h. Can quantum
mechanics multiply large sparse matrices quickly? And the generic answer is yes, it can multiply
large sparse matrices faster than a classical computer, well at least for the Boolean matrix case
and maybe for other cases. There should be a citation here but there's not. But there is a Noon
algorithm to do this, but it's still linear or n to the three halves in the size of the matrix. This
matrix is still exponential so this doesn't help us very much. I mean it helps us a little bit, but it
doesn't solve our problem. Things are still exponential. But it's a little bit inspiring. Maybe if all
we need to do is this matrix vector multiplication maybe we can make things faster. Maybe if
we can use the Hamiltonian as local. Maybe there are games we can play in this context. It's
not clear. It's a little bit unfortunate, right, if this was either the minus ith and not either the
minus tau h that we were working with the world would be much better. Those sort of things
we do know how to multiply fast because these things are unitary and things that are unitary
we can essentially write down as a quantum circuit for large classes of Hamiltonians, but the
world is not that kind to us, so this is sort of where we are stuck. So we said look, we can't use
quantum mechanics at least sort of in a naïve way to multiply these matrices fast, so could we
go back and use stochasticity? Can we use randomness? Randomness worked for us earlier;
maybe randomness will work for us again. Let's again think about classical physics. Let's think
about using randomness to help us out. So what would we need in order to use randomness?
What did we need last time? We needed to build some little black box that samples from
essentially the ground state or the ground state squared. And what do we have with quick
access to these matrix elements, the matrix elements that use the minus tau h. Okay, so what
is our game? We have this vector that we started with, some arbitrary guess and we want to
multiply it by this matrix and this matrix and this matrix. We can't do that because these
matrices are too large, so let's just choose from a random path through this multiplication and
that gives us some weight for some matrix element in the end vector. And then let's give this
random path that we've chosen a weight. Its weight is that matrix element times that matrix
element times this matrix element times this matrix element. Okay, so we have a bunch of
paths. Each path has a weight associated with it and then let's choose the path with probability
proportional to the weight. This sounds like a good idea. In fact, there is a reasonably
straightforward way to do this using Markoff chains, write down the path, change the path a
little bit, accept or reject and we do this over and over again. We could do this using Diffusion
Monte Carlo. There are lots of ways that you can play this game. So this looks very good for us,
right? This looks like again we have somehow used randomness to multiply these matrices fast,
so we have some probability of a path that is proportional to the product of these matrix
elements and if we want to compute something we just sample these paths essentially and we
measure some observable. Okay, it looks like we are done. It looks like we can declare success
and not worry about the quantum computing at all. But there's something that has gone wrong
here, something that I sort of slipped under the rug that is particularly important. So I said in
this statement look we have these paths and we want to choose the paths with the probability
proportional to the product of the elements. And that's all well and good as long as the
product of the elements look a lot like probability, and that happens some of the time and in
those cases we are lucky. But in fermionic simulations this will not be the case and it will not be
the case in a very severe way. These paths, sometimes these weights will sometimes be
negative, and it's very painful to choose something with the probability of a negative number,
so that's bad. So this is canonically what's called the sign problem and this is one manifestation
of the sign problem. So we have this nice stochastic approach and it fails here. You say alright,
maybe we can just sort of tweak it a little bit and get back what we had before. So instead of
sampling with the probability of elements, let's just sample the paths with the probability of a
product with the magnitude of the elements. It seems like a reasonable approach, so we have
these positive paths and these negative paths and we’ll just take the magnitudes. Then you've
got to clean something up, right, because that's not what you meant to be doing, so instead of
having just the observable here, you'll also have to tack on the sign. So I'm hiding something a
little bit with the denominator for experts, but essentially this is what you would have to do.
And you say okay great, I've fixed my problem and we are done again. No more worrying about
quantum computing. No more worrying about anything else. We know how to simulate things
classically and formerly in some sense this is a valid algorithm. This will work. This will give you
the right answer, but there is still a problem and the problem is that there is some variance to
this object. It doesn't help if I stand in front of something. There's some variance to that object
and the variance is exponential with respect to the meaning of that value. So that's bad and
this is what's called the sign problem, so if I was willing to run for exponential amount of time
I'd be fine, but okay, if I'm willing to run an exponential amount of time I could have just
multiplied the matrices to begin with. So this is a problem. So stochasticity does help us in
some cases but this randomness doesn't help us generically. Now there are classical tricks that
you can use to help attenuate this problem somewhat. So one common classical trick is this
idea of a annihilation, so this has been used both in something called Fermion Monte Carlo by
Malkalos [phonetic] and company as well as FC [inaudible] pioneered by [inaudible] and
company and what they say is the following. You have some positive paths. You have some
negative paths and sometimes both a positive and negative path end up in the same spot and
what you want to do when the positive and the negative path end up in the same spot is you
don't want to count it at all. You just want to set that to zero essentially. Okay, that's a nice
idea, and this helps but it only attenuates the sign problem somewhat. At the end of the day,
the deal is that the fact that there are negative paths really kills us. So now let me ask even in
the face of that, I said if you are willing to work hard enough then you can attenuate the sign
problem somewhat with some tricks. How far can you push? And so here's an answer to sort
of how far you can push if you want to get an exact answer. Exact here is sort of in quotes
because almost all of these exact approaches require a little bit of extrapolation, so I'm going to
say exact in quotes for the moment and you can ask what can you do. So this is a scattering of
mainly other people's work, although I've done some of these calculations. So super fluid
helium what are the axes here, right? You have a new sign problem. You can do lots of
particles. The sign problem gets worse and worse and you can do fewer and fewer particles.
And as a rough proxy the more strongly correlated your system is the worst the sign problem is.
This is not a formal statement. This is a rough proxy of how the world looks. So for super fluid
helium it's bosonic. There is no sign problem. You can do a 1000, 10,000 as many particles as
you want for all intents and purposes. So most of these here are the work of the Alave’
[phonetic] group and there they said look, do some sort of electron gas. I think James Spencer
did some of these too. Electron gases, there are two molecules and for weakly correlated
electron gas maybe you can do I don't know maybe 50 particles. For a strongly correlated
electron gas maybe you can do 14 particles. I've done some work on some polarum [phonetic]
problems using these methods and some variants of these methods where we improve them a
bit. There you can essentially do one or two or three particle whole pairs. As you get more
strongly correlated things get bad fast. So this is sort of the manifold of seats that you can do if
you are willing to sort of do things exactly classically. And so just sort of as a summary here and
you might say look, is this really sort of summarize all of what can be done exactly classically?
Well what can you do? You can do sign problem free things and then you can do small systems.
That's really what this graph summarizes. There's other work sort of on independent axes so
we know that if there is low entanglement the work you have to do is sort of exponential and
entanglement and this is a work of O’doul [phonetic] and company. So we know that that's
something that you can do classically and then there's sort of these mashed gates or what
really turns out to be free fermions. This is work of Valiant and company, so this is sort of a box
summarizing the sort of things that you can do classically if the name of the game is I'm only
willing to work with exact answers. So let's step back a second and say look, we said we were in
trouble classically because of this sign problem. Maybe we can go ahead and we can use
quantum computing to somehow deal with this, so we will use quantum computing plus
randomness in sort of this framework. Can that help us all? So what sort of is the most naïve
thing you might imagine trying? The most naïve thing you might imagine trying is to say look,
pretend you can get some quantum circuit that gives you a super position over these paths.
Would this help you? Would this save you somehow? So what would that super position look?
You have some weight on path one and then ended up in position one. You have some weight
on path two and it ended up in position three. You have some weight on path three and also
ended up in position three, but one of those paths was negative and one of those paths was
positive and that's what killed you classically. What killed you classically is that one of these
paths was negative and one of these paths was positive. And let's just say you measure that
quantum state. You measure the dis register. What will happen? Well, you will get R1 with
some probability proportional to alpha 1 squared and that's really what you wanted. But you
will get R3 proportional to alpha 2 squared plus alpha three squared and that's not what you
wanted. What you wanted is you wanted alpha-2 minus alpha three squared, so what killed
you here? Essentially what killed you is you have this memory of this path. You can somehow
forget the path that you took. Then in fact you will get the thing that you want. You would sort
of win. You would get this idea of annihilation that people use classically. They can do
approximately in a classical way. You could actually do exactly in a quantum way. But this is
somehow not obvious how to do it. It's not obvious how in a reversible computation like this
you essentially forget the path. I think it turns out you can do this if the weight of all your
matrix elements are somehow identical, but I think generically this is probably a hard thing.
Nonetheless, if we could do this you might imagine this approach might be cheaper or different
than some of the other approaches that people take. Okay. So this is sort of a Quantum
Quantum Monte Carlo. Now there's lots of literature actually about Quantum Quantum Monte
Carlo, so I should sort of contrast sort of that with what's in the literature. So in the literature
they say look, this is a silly approach. You are still sort of working in the Z basis. You are
working in some electron basis and that's why you have a sign problem to begin with. If you
could just work in the Eigen basis you wouldn't have a sign problem. Okay, so one algorithm
that people hope is sort of a general polynomial time algorithm, so this is originally these
people who wrote this down. And say look, okay, this is a character, but this is the general idea.
Start in a random state. Use phase estimation, collapse into an eigenstate. You collapse into
the wrong thing, reject and go back, otherwise accept with some probability. So this is a
Markov chain, much like this Markov chain, but this works in the Eigen basis and this is working
in the SZ or the electron basis. So this is the advantage of this new sign problem to begin with.
You don't have to worry about it. It has the following disadvantages. Every step you have to do
a phase estimation, so if every phase estimation takes a month, then you need to do, I don't
know, a million steps; that's a long time to wait. The other disadvantage is that here we have
lots of classical experience about how long this chain takes to equilibrate with the mixing time
of this Markov chain. Here we have no real experience at all. I mean if you are sort of keeping
an insulator, sort of a product state, I can argue that it's fast. But generically I don't have much
of a sense for how long these things should take, and so this is the sort of thing I mean when I
say look, lots of people say we have polynomial time algorithms but there's always a hidden
assumption. They hidden assumption here is that the mixing of the Markov chain is actually
fast, which is probably true, but not convincingly true. Okay. So this was my slide before I said
look, polynomial time classically these are the things we can do. What are sort of the games
that we have for polynomial time quantum mechanically? Well this is, we can sort of do
Markov chains in the Eigen basis. That's what I just showed you, quantum Markov chains. We
can do sort of adiabatic paths. This is this game of adiabatic quantum computing. Maybe we
can couple to a heat bath and cool things down. The other thing is really what Jarrod is talking
about is maybe we can start in some place that's really good and then we have large fidelity
over the thing we care about and then we can project and do anything we want. So okay, these
are the sort of games that people know how to play quantumly. I don't know any other games,
but maybe there are some. Now I'm going to pause for a second and I'm going to say look, the
exact part of this talk is over. These are the games one can play if one wants to do things
exactly. Now let's talk about doing things approximately, and in particular, in the spirit of
guessing the right answer. But before we go on I should ask if there are any questions because
this is sort of a natural juncture in this talk.
>>: [inaudible] physical meaning to this [inaudible]?
>> Bryan Clark: To these paths?
>>: Yeah.
>> Bryan Clark: It's not clear that there is a clean physical meaning. In some sense you might
think of this as time evolution in imaginary time meaning that is a perfectly good way of
thinking about it. And so…
>>: You look up the [inaudible]
>> Bryan Clark: Yes [inaudible]
>>: [inaudible] the state of the system. Is there any way [inaudible]
>> Bryan Clark: Well, the matrix elements there is physical meaning to right, but now you have
sort of this product of matrix elements, these paths, and so you lose a lot of physical meaning. I
mean there is physics you can get out of it. For example, if you ramp to a quantum critical point
in imaginary time you can read off critical exponents and things and there are interesting
things, but I wouldn't say it as a general physical picture other than imaginary time path
intervals that I can point to and tell you what this means. Yes?
>>: I like to interject with an open question I think you read into [inaudible] which is so you
reference that there is a quantum algorithm to perform powers of the matrix on the vector and
so there is such an algorithm that people have written down by Horrow and Lloyd [phonetic]
that you could apply an arbitrary function and a matrix to the vector if you only want to sample
the vector like properties of it. But of course the problem is that it succeeds probabilistically
based on the condition number which is related to non-unitarity and the reason that I think this
is an interesting open question is that if you think of every classical linear algebra algorithm that
we have [inaudible] subspace methods, Davidson diagonalization, they all depend on this
projection and renormalization that's inherently non-unitary and I think it's interesting to ask
how you can do modern linear algebra like you were talking about on a quantum computer. It's
just inherently unitary.
>> Bryan Clark: Thanks. Any other comments, questions before I move on? Now I'm going to
talk about sort of the other half of this talk and say look, getting exact answers is hard, so let's
just worry about getting good answers. This is where almost all of the real practical Quantum
Monte Carlo that's done in the world works. So that's what I will talk about. So to zeroith
order, what do we do? We guess the answer. We guess a wave function. So what we do is we
say look, we have some wave function and we are just going to guess what it is. And we know
which one is best because there is a variational principle and the right wave function has the
lowest energy and so we will use that as a proxy for the quality of our guess. In practice work
we really do is write down a large class of possible wave functions and we try to optimize in this
class. Okay, this is very similar to what you are talking about, but this is just what's done
classically. And of course we might ask as an interesting question can we optimize better on a
quantum computer just as sort of an interesting aside. The classically the game is we write
something down and we try to minimize it. The lines have changed here, but okay, and you
might ask what sort of things do we write down. Well, one thing we write down is we say look,
we have some wave function and for free fermions where there's no interaction at all this is just
a state indeterminate. And then one can actually build up sort of a hierarchy of more
sophisticated wave functions in a systematic way but let's not even worry about that. Let's just
say look, most of these wave functions we take this basic object. It's later determined a single
particle orbitals. We address it in some way. So you might write down a Slater determinate
time some two body jastrow factor that addresses it. Or you might say look, instead of doing
one determinate, let's do 100 determinants. You might say add some sort of backflow where
instead of being a single particle orbital of the electron position, it's the single particle orbital of
some quasi-particle, or you might do some BCS wave function, a whole class of wave functions
we can write down. So I think an interesting question is can you build these all on a quantum
computer? I mean just because I can write them down classically, it's not totally obvious to me
although maybe it is totally obvious to someone in the audience, that you can build these on a
classical computer. There is some literature from Ortiz and company and Slater-jastrows and
multi-Slater-jastrows but things like Slater-jastrow backflow I'm much less sure about. As per
the last talk if you could build these on a quantum computer and if you are close then you do
phase estimation and you can reasonably project into the true ground state. Yes [inaudible]?
>>: I have a question. I mean the jastrow is a nonunion tree, so how do you apply, how do you
go about doing the [inaudible]
>> Bryan Clark: I haven't totally wrapped my head around this yet, but there is a long paper by
Ortiz that claims it can be done. So the short answer is there is there's claims in the literature
that it can be done. But I haven't totally wrapped my head around how it's done.
>>: By defining a larger space you can always define non-unitary part of subspace.
>> Bryan Clark: Sure. But it's not obvious to me that sort of arbitrary Slater-jastrows can be
done polynomial -- I mean anything can be done, but that it can be done in polynomial time is
not obvious to me.
>>: [inaudible] I mean [inaudible]
>> Bryan Clark: As long as you project down…
>>: It's the measurement that projects you back down [inaudible]
>>: [inaudible]
>>: That was the concern that I had [inaudible] algorithms they do it by expanding the space
and then measuring down.
>>: It's sort of a funny [inaudible] if you addressed it by looking at say like master equation
dynamics, then of course just like if you're running quantum trajectories simulation you have to
run it multiple times. But you could approximate the dynamics. But this is where you actually, I
guess you, I mean depending on your measurement you throw it away all of the time, so I think
the question is how often do you have to throw it away?
>> Bryan Clark: The other thing I should say is I'm a little bit pessimistic actually about this
generic approach for quantum computing. I sort of think that if I can write down a wave
function where I'm really close to the right answer to begin with anyway and have good fidelity,
I suspect that I can actually get this done classically. And so it just makes me a tad pessimistic in
this thing. On the other hand, okay, it's still a reasonable thing to think about and lots of people
have thought about it in different contexts. Okay. So we can ask sort of practical question.
How far can we push using this kind of approach? This is some work I was involved in. We said
look, let's take a water molecule and this is the exact answer and quantum chemists are very
good at doing small molecules so the quantum chemist answers very close. And if the only
game you're willing to play us you are saying you are just going to write down some variational
wave function, how close can I get? And the wave function we looked at in particular here is
this multi-Slater jastrow. You can get close but not that close. This is sort of a .01 hard tree
error. And this is using, I don't know, 10,000 determinants or a 1000 determinants, so actually
as I take one determinate that is really far away and as I push to just five or six thousand
determinants I get closer and this is sort of how far you can get. I should say there is a
difference here between sort of how quantum chemists push the determinants and how
Quantum Monte Carlo people push determinants because the addition of the jastrow really lets
you use a lot fewer of the determinants then you naïvely might think you could otherwise. So
this is how far you can push something classically with something like just guessing the wave
function. I should sort of say as an interlude that algorithms really matter so this is like a result
that is a year old or something and until very recently the idea of doing Quantum Monte Carlo
10,000 determinants was really kind of a ridiculous thing to try because naïvely you might think
look, if I want to do 10,000 determinants I have to work 10,000 times as hard as doing a single
determinant, and for general determinants it turns out that's true. But we were able to write
down a cute algorithm that actually allows you to do 10,000 determinants with sort of an
additive cost of 10,000 and not a multiplicative cost of 10,000. So this is just an interlude.
Algorithms are important, even classically and we aren't really at what I would say are the limits
of what classical computers can do. So we are having a whole workshop saying look, here's
where we can do now and if we have a quantum computer we just have to beat this, but
there's really a lot of algorithmic development going on classically and so if you're not going to
build your quantum computer for a decade or so, you are really going to have to be shooting
much higher I think then where you are shooting right this second. Yes?
>>: [inaudible] back to your [inaudible], you want to know [inaudible] write something down
[inaudible]
>> Bryan Clark: These are first-class [inaudible].
>>: [inaudible] anyone seriously thinks [inaudible] wave function [inaudible] result [inaudible]
>> Bryan Clark: Well, you could write these all on second quantum notation too. I don't think
the quantization is critical here.
>>: It makes a big difference.
>> Bryan Clark: In practical terms it may make a difference. I've written these down in a first
quantize way but one can break down the equivalent second quantize things. I think this is
more a language than something fundamental.
>>: In practice it would do simulations exactly the same way [inaudible]
>>: But if you [inaudible] write these down [inaudible]
>> Bryan Clark: It becomes very obvious that what?
>>: [inaudible] that those become [inaudible].
>> Bryan Clark: They could re-create them efficiently, okay.
>>: What was that? I couldn't hear.
>> Bryan Clark: I think the statement was that if you could write them down in second quantize
notation you might more likely think that you could create them efficiently on a quantum
computer. Okay. I don't have that intuition, but some people…
>>: So the jastrow becomes an exponential which is diagonal in the qubit base, but it's not a
[inaudible].
>> Bryan Clark: Certainly no more obvious to me although I come from a land where we think
mainly in first quantize notation and sometimes in second, but I think that this is just a
language…
>>: [inaudible]
>> Bryan Clark: Yeah, that's true. So now I've told you about guessing the wave function. I told
you about projecting the ground state. Now actually there is a very [inaudible] in Quantum
Monte Carlo called fixed node Diffusion Monte Carlo that in some sense combines these two
and really this is what's done in practice. I mean everyone who does Quantum Monte Carlo
99% of the time the Quantum Monte Carlo that is done is actually some fixed node Diffusion
Monte Carlo. So you notice I only showed you one example when I said how far can we push
this and the reason I only showed you one example is because no one publishes variational
Monte Carlo results. Everyone publishes Diffusion Monte Carlo results. And so I worked on this
and I have this data, but okay, so let me tell you about what fixed node Diffusion Monte Carlo
is. We talked about projection and we talked about these approximate techniques. So
projection is you just apply stochastically and you have exponential variants. You have to wait a
long time but you just sort of eat that variance anyway long time. And the other approach is
you guess a wave function and the fixed node is really the combination. Fixed node says look,
guess some part of the wave function and do projection on the rest of it. So the part of the
wave function you guess is where the wave function is zero and then it turns out that you can
do projection on the rest of it without a sign problem. So that's a nice feature. So let me
explain that to you in sort of a graphical way. So this is a bit tricky to see with ground seats
because you have to have lots of dimensions in your head, so we will talk about the excited
states because it also works for excited states. So here's a particle in a box. And let's say you
wanted to solve the fermionic first excited state for a particle in a box. You do naïve projection
and what you get is you get a bunch of positive walkers that look like the bosonic solution and a
bunch of negative walkers that look like the bosonic solution and you subtract them and you
get the right answer but it's very small and that's where the sign problem comes in. But now
here is the new game. You guess where the zero is of your answer. So let's say we guess the
right place. We know that that the fermionic solution looks like this. The first excited state
looks like that and we guess where the zero is. The zero should be there and then we solve a
bosonic problem to the left and we solve a bosonic problem to the right. We get the right
answer. Okay, the sign is wrong but we know how to flip the sign. So we get the fermionic
answer. But that presumed I could guess where the zero is. Let's say I screwed up and let's say
I guessed the zero is really here. I solve the bosonic problem to the left and I solve the bosonic
problem to the right and here is the answer that I get. It looks like this. That's not the right
answer and its higher energy than the right answer. You can just see this. This is going to have
much higher energy than these other things, and that's a generic principle. You can just prove
this. You guessed the wrong zeros and you get some answer and that answer has higher
energy. In fact, this wave function that you get out of this procedure is actually a variational
wave function. So an interesting question that I've been thinking about recently that is not at
all obvious, is can we one, build these quote unquote variational wave functions on a quantum
computer. So your game is I have some guess where the zeros are and I want to build a fixed
node diffusion variational wave function. Is that something that I know how to do? Yes?
>>: [inaudible] confined to one dimension?
>> Bryan Clark: No, no only the cartoon is confined to one dimension because my head only
works there.
>>: The cartoon seems to work best because in high dimensions zero is required to [inaudible]
>> Bryan Clark: That's right, so how do you guess? What you do is you guess the best wave
function that you can because that's one way to guess. You take the zeros, so then in some
sense it improves the amplitudes inside the places where there are amplitudes, but the zeros
are fixed to where you put them. There's actually probably something slightly better you can
do and some optimization in some global sense, but in the zeroth order this is what everyone
does. They guess the best wave function they can guess, use those zeros and then clean up the
answer. So that is how you input that information in high dimensions. So now we can ask how
far can we push this. This is what everyone does. So this was the best answer with, I don't
know, 5000 determinants for variational Monte Carlo. Then you use this Diffusion Monte Carlo,
fixed node Diffusion Monte Carlo. You get right down to the best quantum chemistry. So fixed
node Diffusion Monte Carlo with these number of things is competing comfortably with the
best quantum chemistry although there is still a little bit of difference.
>>: [inaudible]
>> Bryan Clark: CCSD with perturbitive triples R1, 2 extrapolated on some basis [inaudible].
This is what the quantum chemists tell me is the gold standard if you are not doing abstract. I
am not a quantum chemist so the quantum chemists may argue amongst themselves that that's
the best thing to do.
>>: I mean how do you know you exact [inaudible] also quantum chemistry, right?
>> Bryan Clark: I don't know where the exact is. I don't remember the exact [inaudible]
[laughter]
>>: The best chemistry you are showing there [inaudible] is…
>> Bryan Clark: And then the exact is extrapolated.
>>: [inaudible] extrapolation.
>> Bryan Clark: Okay. I don't know that that's actually true, but I don't remember off the top of
my head. Anyway, let's say the best Quantum Monte Carlo is close to the best quantum
chemistry. The advantage at least in respect to a quantum computer is twofold. Classically the
advantage of this probably scales better, the Quantum Monte Carlo. The advantage with
respect to a quantum computer is this is some variational wave function so you might imagine
being able to dial it into a quantum computer, whereas the best quantum chemistry as I
understand doing some perturbitive triples isn't really a wave function and so it's not clear what
you do with that object with respect to sort of building things in the quantum computation,
although maybe there is some natural thing. Okay. So that was a water molecule but we've
done lots and lots of molecules so if you are a quantum chemist this is the G1 set and we are
down to sort of an error of I don't know, two or three Miller hard tree from the true answer on
many, many molecules, smallish but not all that smallish molecules. So this is sort of what the
best approximate things can do, and then you say look, and you push to bigger things. So this
isn't my work. This is Claudia [inaudible] work and company and you say look, this is [inaudible]
is important for something in your eye and now we are sort of at the point where there is really
nothing to compare it to. So I said before for biggish systems Quantum Monte Carlo is kind of
the de facto gold standard. We are now kind of at the point where I don't really know how to
tell you how good this calculation is because it gives you some answer and what are you going
to compare it to. It's the de facto gold standard. It's not exact, but it's reasonably good. You
can compare it to experiment. There are errors on experiment. There are errors on these wave
functions. This is actually some gap, but okay, this is sort of the order that we can do this in. So
we can do sort of 1000 electrons and pushing hard to sort of on the order of 10,000 electrons.
Here is something I was involved in, although the main offer on this is John Grigley [phonetic].
So we said look instead of doing a single water molecule let's do a bulk system. And so we will
do bulk periodic water. We will do 32 or 64 water molecules. We will take some snapshots and
we will compute properties, ground state energies et cetera. Again, I don't know how to tell
you how good we got here, because this is the de facto gold standard. And you said what's the
best you can do. And I said go do some Quantum Monte Carlo which is what we did. And
actually we were using, this is a benchmark for other things, so if we compare this against DFT,
DFT is off on the order of sort of 20 Kelvin per molecule, which is sort of disastrous if you
actually care about what's actually going on with water. Again, this is a graph that shows how
bad DFT is. So this is why I call it a better DFT. In some sense there is a number of advantages
compared to DFT. One of them is that it's more accurate and the second one is in DFT you have
1000 functionals and who knows which one is better and which one is worse. Here you at least
have some metric to decide which is better, which is if your energy is lower it's better. Okay.
But again, I told you a couple of things. I said look, here the kind of system sizes that people do,
sort of as bread-and-butter Quantum Monte Carlo, but I can't tell you how accurate they are, so
how should we quantify this. Can I at least give you some numbers to sort of give you a feel for
what kind of ballpark we need to be in, for quantum computers to be quote unquote better. So
if we just think about the energy of a molecule or something of some [inaudible] Hamiltonian,
most of that energy is just uncorrelated, which is easy to get for free Hartree-Fock or
something. There is some little sliver of that, 3% or so that is sort of correlated energy and
Quantum Monte Carlo is able to get sort of 95% of that remaining correlation energy back.
That's sort of the order of magnitude that it can get back, so it's really this remaining sort of
slice of the pie, this remaining 5% of the correlation energy that we need to get back. So if
you're going to say look I'm on a quantum computer and I have some good approximation
algorithm on it, what that approximation algorithm needs to do is be eating into this sort of
final remaining 5% of the correlation energy. Now there's lots of interesting physics that lives in
that final remaining 5%, magnetism, superconductivity lots of things live there so it's important
to get it, but this is sort of what we are talking about, 3% of 5% of the energy that we really
have to be sitting at before you can really sort of convincingly say you are doing better than
today's state-of-the-art. Okay, yes?
>>: [inaudible] what is modern Monte Carlo tell us now about [inaudible]? What are we, can I
look at these iron sulfur complexes and get [inaudible]?
>> Bryan Clark: So that's a good question, so how do people do these catalysts? I'm not a
quantum chemist so my understanding is you have some set of nuclei around and you push
them with forces in some carpinello way. Is that an accurate understanding of how this game is
played? So this is my understanding. So then you might say well, why haven't we all done this
with Quantum Monte Carlo and the answer is the following. For each step of these pushing
around with the nuclei, a carpinello calculation takes a couple of seconds, 30 seconds, 12
seconds, something like that. And that's because a single DFT run is fast. For each step you
have to do a DFT. So Quantum Monte Carlo scales the same way, parallelizes much better, all
kinds of great things but is two orders of magnitude slower for each step. On top of that
Quantum Monte Carlo getting force is a little bit difficult, sort of the state-of-the-art. We are
sort of on the border of being able to do this in an accurate way but it's still noisy and we have
to worry about electrodynamics of noisy forces. So we say that this is the sort of thing that I
imagine Quantum Monte Carlo methods to begin being applied to in the not-too-distant future,
but is not at the point where there is sort of new code that you can download like there is in
DFT and you just hit a button and get results in some reasonable way. This is still a point where
there is methodological developments that seem tractable, seem doable but there are is still
methodological developments there in the future. But that's the right thing to do. I mean
when I heard this talk at I said okay, we did DFT and we have no idea what's going on. I mean
the first thing any Quantum Monte Carlo person thinks is okay, we should be doing Quantum
Monte Carlo. Yes?
>>: [inaudible] because you don't in DFT [inaudible] the point of trying to make. Because you
don't need to know the dynamics all you are looking for [inaudible] the surface and you get
them quicker than pushing the nuclei around.
>> Bryan Clark: Okay.
>>: You get information from them…
>> Bryan Clark: You still need to do some sort of gradients or something in that. So it's still not
quite that bad but it still a bit painful. I mean we are talking two orders of magnitude, right, so
you're doing a complicated calculation and now I'm telling you it's going to be two orders of
magnitude slower. Okay. So in conclusion we can ask where does this leave us. So classically
these are the sort of things that we can do exactly, so for small systems for sort of weakly
correlated, I don't know. We're talking may be 50 electrons, 20 electrons, something like that.
If we want to do approximation well, then we are really talking about 1000, 10,000 electrons
and actually in a decade many more than that. Things scale polynomially and we are sort of
eating 95% of the correlation energy. The remaining 5% okay there's active development and
better approximations but we are not there yet. This is where all of the Quantum Monte Carlo
is. There's many interesting systems. As far as quantum exact methods, we already discussed
sort of that IC is sort of the only game in town right this second. So it would be nice to sort of
expand this list and say look, can we do something else. And these are sort of the let's say
questions that I threw out there and I said look, could we for example, do fast matrix
multiplication. Can we figure out in this random method how to forget the paths? Can we
build fixed node wave functions? These are sort of some of the questions, some of the
discussion points I think might be good starting points for saying look, can we build better ways
of doing quantum simulations on quantum computers. Right, that's it?
>>: Thank you [applause].
>>: Questions?
>>: [inaudible] error [inaudible]?
>> Bryan Clark: For the exact methods?
>>: [inaudible]
>> Bryan Clark: I mean the error at this point is statistical. There is no systematic error. They
are sort of negligiblish systematic error.
>>: [inaudible] Monte Carlo [inaudible] error [inaudible]
>> Bryan Clark: Yeah, I mean you're asking for the statistical error? And I should have that but I
don't remember off the top of my head, but I could look it up. I mean it's reasonably small.
Yeah, it's reasonably small. I mean right, you can solve anything with big error, so when I say
around 50 electrons what I mean is some reasonable chemical accuracyish, smallish statistical
error.
>>: Okay. Thank you. Let's go to lunch now. [applause]