97 research outputs found
Obstructions To Classically Simulating The Quantum Adiabatic Algorithm
We consider the adiabatic quantum algorithm for systems with "no sign
problem", such as the transverse field Ising mode, and analyze the
equilibration time for quantum Monte Carlo (QMC) on these systems. We ask: if
the spectral gap is only inverse polynomially small, will equilibration methods
based on slowly changing the Hamiltonian parameters in the QMC simulation
succeed in a polynomial time? We show that this is not true, by constructing
counter-examples. Some examples are Hamiltonians where the space of
configurations where the wavefunction has non-negligible amplitude has a
nontrivial fundamental group, causing the space of trajectories in imaginary
time to break into disconnected components, with only negligible probability
outside these components. For the simplest example we give with an abelian
fundamental group, QMC does not equilibrate but still solves the optimization
problem. More severe effects leading to failure to solve the optimization can
occur when the fundamental group is a free group on two generators. Other
examples where QMC fails have a trivial fundamental group, but still use ideas
from topology relating group presentations to simplicial complexes. We define
gadgets to realize these Hamiltonians as the effective low-energy dynamics of a
transverse field Ising model. We present some analytic results on equilibration
times which may be of some independent interest in the theory of equilibration
of Markov chains. Conversely, we show that a small spectral gap implies slow
equilibration at low temperature for some initial conditions and for a natural
choice of local QMC updates.Comment: 18 pages plus appendix. Note on latex compilation: this should
compile with pdflatex or with latex followed by dvipdfm, but compiling with
latex followed by dvips may cause the figures in the appendix to render
incorrectly; v2: added reference
Diffusion Monte Carlo approach versus adiabatic computation for local Hamiltonians
Most research regarding quantum adiabatic optimization has focused on
stoquastic Hamiltonians, whose ground states can be expressed with only real,
nonnegative amplitudes. This raises the question of whether classical Monte
Carlo algorithms can efficiently simulate quantum adiabatic optimization with
stoquastic Hamiltonians. Recent results have given counterexamples in which
path integral and diffusion Monte Carlo fail to do so. However, most adiabatic
optimization algorithms, such as for solving MAX-k-SAT problems, use k-local
Hamiltonians, whereas our previous counterexample for diffusion Monte Carlo
involved n-body interactions. Here we present a new 6-local counterexample
which demonstrates that even for these local Hamiltonians there are cases where
diffusion Monte Carlo cannot efficiently simulate quantum adiabatic
optimization. Furthermore, we perform empirical testing of diffusion Monte
Carlo on a standard well-studied class of permutation-symmetric tunneling
problems and similarly find large advantages for quantum optimization over
diffusion Monte Carlo.Comment: 7 pages, 5 figures, updated organization, typos, journal reference
added (results unchanged
Adiabatic optimization versus diffusion Monte Carlo
Most experimental and theoretical studies of adiabatic optimization use
stoquastic Hamiltonians, whose ground states are expressible using only real
nonnegative amplitudes. This raises a question as to whether classical Monte
Carlo methods can simulate stoquastic adiabatic algorithms with polynomial
overhead. Here, we analyze diffusion Monte Carlo algorithms. We argue that,
based on differences between L1 and L2 normalized states, these algorithms
suffer from certain obstructions preventing them from efficiently simulating
stoquastic adiabatic evolution in generality. In practice however, we obtain
good performance by introducing a method that we call Substochastic Monte
Carlo. In fact, our simulations are good classical optimization algorithms in
their own right, competitive with the best previously known heuristic solvers
for MAX-k-SAT at k=2,3,4.Comment: 13 pages plus appendices. Published versio
Polynomial Time Algorithms for Estimating Spectra of Adiabatic Hamiltonians
Much research regarding quantum adiabatic optimization has focused on
stoquastic Hamiltonians with Hamming symmetric potentials, such as the well
studied "spike" example. Due to the large amount of symmetry in these
potentials such problems are readily open to analysis both analytically and
computationally. However, more realistic potentials do not have such a high
degree of symmetry and may have many local minima. Here we present a somewhat
more realistic class of problems consisting of many individually Hamming
symmetric potential wells. For two or three such wells we demonstrate that such
a problem can be solved exactly in time polynomial in the number of qubits and
wells. For greater than three wells, we present a tight binding approach with
which to efficiently analyze the performance of such Hamiltonians in an
adiabatic computation. We provide several basic examples designed to highlight
the usefulness of this toy model and to give insight into using the tight
binding approach to examining it, including: (1) adiabatic unstructured search
with a transverse field driver and a prior guess to the marked item and (2) a
scheme for adiabatically simulating the ground states of small collections of
strongly interacting spins, with an explicit demonstration for an Ising model
Hamiltonian.Comment: 12 pages, 9 figures, v3: update to be consistent with Phys. Rev. A
versio
Quantum Supremacy through the Quantum Approximate Optimization Algorithm
The Quantum Approximate Optimization Algorithm (QAOA) is designed to run on a
gate model quantum computer and has shallow depth. It takes as input a
combinatorial optimization problem and outputs a string that satisfies a high
fraction of the maximum number of clauses that can be satisfied. For certain
problems the lowest depth version of the QAOA has provable performance
guarantees although there exist classical algorithms that have better
guarantees. Here we argue that beyond its possible computational value the QAOA
can exhibit a form of Quantum Supremacy in that, based on reasonable complexity
theoretic assumptions, the output distribution of even the lowest depth version
cannot be efficiently simulated on any classical device. We contrast this with
the case of sampling from the output of a quantum computer running the Quantum
Adiabatic Algorithm (QADI) with the restriction that the Hamiltonian that
governs the evolution is gapped and stoquastic. Here we show that there is an
oracle that would allow sampling from the QADI but even with this oracle, if
one could efficiently classically sample from the output of the QAOA, the
Polynomial Hierarchy would collapse. This suggests that the QAOA is an
excellent candidate to run on near term quantum computers not only because it
may be of use for optimization but also because of its potential as a route to
establishing quantum supremacy.Comment: 23 pages. v2 fixes bug in section 4. Results unchange
The Power of Adiabatic Quantum Computation with No Sign Problem
We show a superpolynomial oracle separation between the power of adiabatic
quantum computation with no sign problem and the power of classical
computation.Comment: 22 pages, 2 figures; v2 final version in Quantu
Scaling analysis and instantons for thermally-assisted tunneling and Quantum Monte Carlo simulations
We develop an instantonic calculus to derive an analytical expression for the
thermally-assisted tunneling decay rate of a metastable state in a fully
connected quantum spin model. The tunneling decay problem can be mapped onto
the Kramers escape problem of a classical random dynamical field. This
dynamical field is simulated efficiently by path integral Quantum Monte Carlo
(QMC). We show analytically that the exponential scaling with the number of
spins of the thermally-assisted quantum tunneling rate and the escape rate of
the QMC process are identical. We relate this effect to the existence of a
dominant instantonic tunneling path. The instanton trajectory is described by
nonlinear dynamical mean-field theory equations for a single site magnetization
vector, which we solve exactly. Finally, we derive scaling relations for the
"spiky" barrier shape when the spin tunnelling and QMC rates scale polynomially
with the number of spins while a purely classical over-the-barrier
activation rate scales exponentially with .Comment: 15 pages, 4 figures, 45 reference
Path-Integral Quantum Monte Carlo simulation with Open-Boundary Conditions
The tunneling decay event of a metastable state in a fully connected quantum
spin model can be simulated efficiently by path integral quantum Monte Carlo
(QMC) [Isakov , Phys. Rev. Lett. , 180402 (2016).]. This is
because the exponential scaling with the number of spins of the
thermally-assisted quantum tunneling rate and the Kramers escape rate of QMC
are identical [Jiang , Phys. Rev. A , 012322 (2017).], a
result of a dominant instantonic tunneling path. In Ref. [1], it was also
conjectured that the escape rate in open-boundary QMC is quadratically larger
than that of conventional periodic-boundary QMC, therefore, open-boundary QMC
might be used as a powerful tool to solve combinatorial optimization problems.
The intuition behind this conjecture is that the action of the instanton in
open-boundary QMC is a half of that in periodic-boundary QMC. Here, we show
that this simple intuition---although very useful in interpreting some
numerical results---deviates from the actual situation in several ways. Using a
fully connected quantum spin model, we derive a set of conditions on the
positions and momenta of the endpoints of the instanton, which remove the extra
degrees of freedom due to open boundaries. In comparison, the half-instanton
conjecture incorrectly sets the momenta at the endpoints to zero. We also found
that the instantons in open-boundary QMC correspond to quantum tunneling events
in the symmetric subspace (maximum total angular momentum) at all temperatures,
whereas the instantons in periodic-boundary QMC typically lie in subspaces with
lower total angular momenta at finite temperatures. This leads to a lesser than
quadratic speedup at finite temperatures. We also outline the generalization of
the instantonic tunneling method to many-qubit systems without permutation
symmetry using spin-coherent-state path integrals.Comment: 10 page
Effective gaps are not effective: quasipolynomial classical simulation of obstructed stoquastic Hamiltonians
All known examples confirming the possibility of an exponential separation
between classical simulation algorithms and stoquastic adiabatic quantum
computing (AQC) exploit symmetries that constrain adiabatic dynamics to
effective, symmetric subspaces. The symmetries produce large effective
eigenvalue gaps, which in turn make adiabatic computation efficient. We present
a classical algorithm to efficiently sample from the effective subspace of a
-local stoquastic Hamiltonian , without a priori knowledge of its
symmetries (or near-symmetries). Our algorithm maps any -local Hamiltonian
to a graph with
where is the number of qubits. Given the well-known result of Babai, we
exploit graph isomorphism to study the automorphisms of and arrive at an
algorithm quasi-polynomial in for producing samples from the
effective subspace eigenstates of . Our results rule out exponential
separations between stoquastic AQC and classical computation that arise from
hidden symmetries in -local Hamiltonians. Furthermore, our graph
representation of is not limited to stoquastic Hamiltonians and may rule
out corresponding obstructions in non-stoquastic cases, or be useful in
studying additional properties of -local Hamiltonians.Comment: 9 pages, 5 figures, added footnotes (v2
Prospects for Quantum Enhancement with Diabatic Quantum Annealing
We assess the prospects for algorithms within the general framework of
quantum annealing (QA) to achieve a quantum speedup relative to classical state
of the art methods in combinatorial optimization and related sampling tasks. We
argue for continued exploration and interest in the QA framework on the basis
that improved coherence times and control capabilities will enable the
near-term exploration of several heuristic quantum optimization algorithms that
have been introduced in the literature. These continuous-time Hamiltonian
computation algorithms rely on control protocols that are more advanced than
those in traditional ground-state QA, while still being considerably simpler
than those used in gate-model implementations. The inclusion of coherent
diabatic transitions to excited states results in a generalization called
diabatic quantum annealing (DQA), which we argue for as the most promising
route to quantum enhancement within this framework. Other promising variants of
traditional QA include reverse annealing and continuous-time quantum walks, as
well as analog analogues of parameterized quantum circuit ansatzes for machine
learning. Most of these algorithms have no known (or likely to be discovered)
efficient classical simulations, and in many cases have promising (but limited)
early signs for the possibility of quantum speedups, making them worthy of
further investigation with quantum hardware in the intermediate-scale regime.
We argue that all of these protocols can be explored in a state-of-the-art
manner by embracing the full range of novel out-of-equilibrium quantum dynamics
generated by time-dependent effective transverse-field Ising Hamiltonians that
can be natively implemented by, e.g., inductively-coupled flux qubits, both
existing and projected at application scale.Comment: A perspective/review. 26 pages, 6 figure
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