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 k-local stoquastic Hamiltonian H, without a priori knowledge of its symmetries (or near-symmetries). Our algorithm maps any k-local Hamiltonian to a graph G=(V,E) with |V|=O(poly(n)) where n is the number of qubits. Given the well-known result of Babai, we exploit graph isomorphism to study the automorphisms of G and arrive at an algorithm quasi-polynomial in |V| for producing samples from the effective subspace eigenstates of H. Our results rule out exponential separations between stoquastic AQC and classical computation that arise from hidden symmetries in k-local Hamiltonians. Furthermore, our graph representation of H is not limited to stoquastic Hamiltonians and may rule out corresponding obstructions in non-stoquastic cases, or be useful in studying additional properties of k-local Hamiltonians.

%8 4/21/2020 %G eng %U https://arxiv.org/abs/2004.08681 %0 Journal Article %D 2019 %T Polynomial Time Algorithms for Estimating Spectra of Adiabatic Hamiltonians %A Jacob Bringewatt %A William Dorland %A Stephen P. Jordan %XMuch 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.

%8 05/25/2019 %G eng %U https://arxiv.org/abs/1905.07461 %0 Journal Article %J Physical Review A %D 2018 %T Diffusion Monte Carlo Versus Adiabatic Computation for Local Hamiltonians %A Jacob Bringewatt %A William Dorland %A Stephen P. Jordan %A Alan Mink %XMost 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.

%B Physical Review A %V 97 %P 022323 %8 2018/02/15 %G eng %U https://journals.aps.org/pra/abstract/10.1103/PhysRevA.97.022323 %N 2 %R 10.1103/PhysRevA.97.022323