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ZHANG JIANG: So hello, everyone.
I'm Zhang Jiang from NASA Ames.
Today I'm going to talk about the instanton method in quantum
annealing, thermal assisted tunneling versus quantum Monte
Carlo simulation.
So this is a collaboration with Vadim Smelyanskiy at Google,
who is an expert in quantum tunneling and transport,
and Sergio Boixo, Hartmut Neven, and Guglielmo Mazzola,
Matthias Troyer at ETH Zurich, and also
Sergei Isakov at Google Zurich.
So it has been found that there exists an analogy between
quantum tunneling and the classical escape-over-a-barrier
problem.
So our central question for us to ask
is, can thermal activation in quantum Monte Carlo simulate
quantum tunneling in quantum annealing?
So here is a brief introduction to what
is quantum Monte Carlo, especially
we use the path-integral of Monte Carlo.
In the left side, it is the quantum system.
The red dots are quantum qubits.
And the arrows are the local field acting at the quantum--
AUDIENCE: [INAUDIBLE]
ZHANG JIANG: Oh, that's weird.
Yeah.
This works.
OK.
So this is the thermal activation
versus quantum tunneling.
OK.
On the left side, this is the quantum system
we are considering.
So each being coupled to every other spins.
And the red dots represent quantum spins.
These yellow arrows are local fields
that are acting on these quantum spins as transverse field.
So one way to map this quantum system to a classical system
is to do trotterization.
Here, I plot three of the replicas,
but you can have many replicas.
So in the classical system, we have these blue dots
representing classical spins.
And the classic spins are linked to each other
with the same potential energy as the quantum spins.
However, there is no transverse field acting
on the classical spin, instead we
have this thermal magnetic coupling
between these classical spins for the corresponding spins
in each replicas.
So why we have this kind of map?
So the reason we consider that is because we can't always
map the partition function of a quantum system
to a classical system.
So to do that, we need to do trotterization
and to put this k as the kinetic energy of the quantum system
and this u as a potential term.
And after we do the trotterization,
we got another potential term for the classical space
where the adjacent replicas are coupled
to each other by this term K beta.
And we call this the classical Hamiltonian, HQMC.
This classical Hamiltonian depend
on the inverse temperature beta.
So mapping from the quantum to classical system is easy.
We have this K and U and we map of this quantum Hamiltonian
to the classical Hamiltonian.
So our question becomes to ask whether our quantum tunneling--
did it described by the quantum Hamiltonian,
whether it can be simulated by the Kramers escape event
within this classic Hamiltonian, HQMC beta x, where
x denotes all the configurations of these replicas.
So this is a plot of Kramers escape event.
Originally your state sits in this local minimum.
And because of film activation, it
goes to the transition point, which is also
called the saddle point.
And then, just go to the global minimum with high probability.
So here, I state the main results
we have found in this research.
First, we find that the saddle point of the quantum Monte
Carlo Hamiltonian, which is the classical Hamiltonian I just
described, is an instanton in the order parameter space.
So an instanton is just a solid [INAUDIBLE] in imaginary time.
And also we find that the exponential scaling
of the escape rate in the quantum Monte Carlo at a given
temperature 1 over beta equals to that
of the quantum tunneling rate in quantum annealing.
So here are the archived versions of our work.
So to describe this correspondence
we need to describe the system of-- and the consideration.
So here, sigma represents the configurations
of the classical spins of many replicas.
So we have n spins couples to each other.
And the coupling is a homogeneous coupling,
so it only depends on the [? Hamming ?]
weight of the classical spin.
And this indices, tau, represents the replicas.
So if you have the trace condition,
then we have the periodic boundary condition.
The reconfiguration at time 0 equal to the configuration
at time beta.
So beta is just the imaginary time corresponding
to the temperature 1 over t.
So the probability of a state vector
of this certain configuration is given by the bottom
at this distribution, where e is the energy
of that configuration.
And this e equals to this expression.
So in the first integral that g m tau is our potential energy.
So g is-- in a function we can choose and tau
is the sum of the spin sigma J.
So this potential energy is [? Hamming ?] symmetric
plus some term from the kinetic energy, the transverse field
of the quantum Hamiltonian.
And kappa just accounts the number
of kinks in this sigma j tau.
So the number that the sigma j tau changes its sign.
And to tackle with this problem we introduce
the other parameter m tau, which is
the sum of the classical space, sigma j.
So the free energy in the other parameter space
have been given by Bapst and Semerjian.
It has this kind of nice form, where
this integral is evaluated.
The second term is kind of tricky.
So this v is the trace of a non-linear propagator--
a non-linear two-dimensional propagator, because this lambda
tau equals to g prime, where g is the potential energy,
the prime is the first derivative of that potential.
So this is a non-linear propagator of this Hamiltonian.
So this is quite tricky to solve.
The way we tackle that is we map this non-linear propagator
to a magnetic field.
And the evolution of the expectation value of sigma x,
sigma y, and sigma z can be mapped to-- a spin-1/2 particle
evolving under the magnetic field.
So in that way we can solve the saddle point
of the free energy.
OK.
So even numerics the paths are sampled
by using the Monte Carlo methods,
for example, Metropolis or [INAUDIBLE].
And the explanation of scaling is
given by this difference in the free energy, delta F.
And the delta F is the free energy
add to the transition point minus the free energy
add to the local minimum.
And the transition point as we show
is given by the instanton solution.
So [INAUDIBLE] you would think that because of the replica's
symmetric-- so the transition point would also be symmetric
so it would be a static point, but we
find that it is real-- counterintuitive.
This is kind of like replica symmetry breaking.
So you have this non-trivial shape of the other parameter
in the space of the replicas.
And the transition state is given by this equation,
where you-- where the derivative of the free energy with respect
to the other parameters and tau vanishes.
And using the method I just talked about mapping this--
mapping the non-linear equation into a spin-1/2 particle
evolving in the magnetic field.
We have been able to derive the instanton
equation of this transition point, which is this equation.
And here, there is a free parameter l.
And this l can be determined by the self-consistent condition,
which I'm not going to talk about here.
And interestingly, we also derive this equation
with the WKB method, where the rate of quantum tunneling
can be evaluated.
And this parameter l has a physical meaning there.
This l is the-- this parameter l is just the total spin quantum
number in the WKB approach, where
your quantum tunneling happens.
So here are our results.
We calculated the scaling of the tunneling coefficient.
We compare it with the numerics, and they can form pretty well
for the Curie Weiss model.
And using the formalism, we developed
that we can analyze some simple model
like this spike Hamiltonian.
And we also derive similar results
as talked about before by [INAUDIBLE] and Lucas
and [INAUDIBLE].
OK.
So here, I want to talk about some new research we've
done recently.
It's about the open condition quantum Monte Carlo.
So the advantage is you have high efficiency,
you got a potential quadratic speedup
over the periodic boundary conditions.
We can have extra control by adding pinning field
on edge spins.
And we also have a nice formalism
to evaluate the free energy for finite number of replicas,
not necessarily a large [INAUDIBLE] limit.
But it also has some disadvantages,
like it has edge effect.
It has extra entropic factors.
And it has unphysical correlation functions,
which not necessarily affect our quantum optimization result.
So what we find is instead of having full instanton,
we have a half instanton in sort of
in the open-boundary condition.
And what's interesting is in the open-boundary condition,
the quantum tunneling always happens
in total angular momentum space equal to the maximum value.
So this could have brought some speed up
with this interesting effect.
And also it could be a quadratically faster
than quantum annealing.
Let me just skip this.
OK.
So this is the result we find for the initial conditions
for this open-boundary instantons.
Because now you have this uncertainty
of the initial position and the initial velocity.
So the instantons are much harder
to evaluate than the closed boundary condition.
But we find it can be derived using these boundary
conditions.
And also we've used the WKB approach
to compare with the quantum Monte Carlo calculations.
So here is the partition function
for the open-boundary condition case.
Instead of having a trace, we have this expectation value
over the state s, where s is the even superposition
in the computational basis.
So because of this s term, here, we
can insert two of the basis-- and 1 and 2.
And that will give us an extra entropic turn
for the open-boundary condition.
We calculate the free energy for the instanton solution
for the-- using the WKB approach, which
equals to the free energy at an instanton solution using
the quantum Monte Carlo approach or meanfield approach.
And, here, this A is the action of the instanton.
And this e is the energy of the instanton.
And this add to entropic factor because of using
the open-boundary condition.
And the same endpoint conditions can also
be derived using this equation.
So here, comes to the conclusion.
So we have showed for a class of problems
with homogeneous coupling or [? Hamming ?] symmetric cost
function, the saddle point of the quantum Monte Carlo
corresponds exactly to the instanton solution.
And we find that the free energy difference at the saddle point
and at the local minimum gives the explanation of scaling
of the quantum Monte Carlo.
So the conclusion is quantum Monte Carlo can simulate
quantum tunneling for all set-- all [? Hamming ?]
symmetric cost functions.
So some future work, we will definitely
want to calculate the prefactor of the quantum Monte Carlo
and the quantum annealing to compare it
with the recent Google result, where you have
this 10 to the eighth speed-up.
And also, we are interested to generalize this instanton
calculus to more general problems like the spin glass
problem by introducing more of the parameter like the replica
symmetry breaking parameter.
And we also want to extend this result to known stoquastic
Hamiltonian.
OK.
Here is my talk.
Thanks for you attention.
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SPEAKER: Any questions?
AUDIENCE: Do you have some ideas in mind
for how you might extend to the non-stoquastic case?
ZHANG JIANG: Yes, I have.
So actually it's quite easy to do
in those stochastic evolution because your Hamiltonian
is [? Hamming ?] symmetric.
So you can add some x terms in your potential term.
If it is [? Hamming ?] symmetric you can.
AUDIENCE: OK, thanks.
SPEAKER: Other questions?
OK, let's thank all the speakers of this session.
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