A quantum–quantum Metropolis algorithm

Abstract

The classical Metropolis sampling method is a cornerstone of many statistical modeling applications that range from physics, chemistry, and biology to economics. This method is particularly suitable for sampling the thermal distributions of classical systems. The challenge of extending this method to the simulation of arbitrary quantum systems is that, in general, eigenstates of quantum Hamiltonians cannot be obtained efficiently with a classical computer. However, this challenge can be overcome by quantum computers.Here, we present a quantum algorithm which fully generalizes the classical Metropolis algorithm to the quantum domain. The meaning of quantum generalization is twofold: The proposed algorithm is not only applicable to both classical and quantum systems, but also offers a quantum speedup relative to the classical counterpart. Furthermore, unlike the classical method of quantum Monte Carlo, this quantum algorithm does not suffer from the negative-sign problem associated with fermionic systems. Applications of this algorithm include the study of low-temperature properties of quantum systems, such as the Hubbard model, and preparing the thermal states of sizable molecules to simulate, for example, chemical reactions at an arbitrary temperature.Metropolis method | statistical physics | quantum computing | quantum simulation | simulated annealing I nteracting many-body problems cover a wide range of applications in many areas in science and technology. These are best exemplified by the Ising spin model, which was originally developed for explaining ferromagnetism and its associated phase transitions (1). The Ising model was later found to be related to many practical applications, for example, error-correcting codes, image restoration, associative memory, and optimization problems (see, e.g., ref. 2). However, there is no efficient method for finding solutions to many-body problems in general. For example, the problem of finding the ground state of the Ising model with arbitrary couplings is known to be a nondeterministic polynomial (NP)-complete problem, meaning that, if an efficient algorithm for solving the Ising model exists, then all of the problems in the class of NP, for example the graph isomorphism problem and some variants of the traveling salesman problem, can be solved efficiently as well (see, e.g., ref.3).The most fundamental challenge for solving many-body problems is that they generally require an exponentially large amount of spatial and temporal computing resources to find the solutions as the system size increases (4). Although no universal method for solving general many-body problems has been found, powerful classical methods such as Markov-chain Monte Carlo (MCMC) or quantum Monte Carlo (QMC) have been invented and proven to be successful in many applications (see, e.g., ref. 5). These methods, however, have certain limitations. For example, the running time of MCMC scales as Oð1∕δÞ (6), where δ is the gap of the transition matrix. For problems such as spin glasses (7) where δ is vani...

Quantum optimization using variational algorithms on near-term quantum devices

“…This can in principle lead to very short-depth circuits which ideally run in a time that is shorter than the coherence time of the quantum computer. The same variational quantum eigensolver can be applied to other physical systems in condensed matter such as the Fermi-Hubbard model [2,45,12,17,46,47,48] and spin systems [49,50,51,52].…”

Adaptive Quantum Simulated Annealing for Bayesian Inference and Estimating Partition Functions

“…An approach by [27] that relies on Metropolis sampling generalizes qsampling to quantum Hamiltonians, but it likewise scales like O(1/δ) in the spectral gap. Another approach, that of quantum simulated annealing (QSA) [22,23,30,31], the approach that we will employ, relies on qsampling the stationary distributions of a series of intermediate Markov chains. Successive stationary distributions satisfy a "slow-varying condition" | Π i |Π i+1 | 2 ≥ const, which allows these algorithms to bound the dependence on min x Π(x) while preserving the O(1/ √ δ) square root scaling in the spectral gap.…”

Untitled

“…For any given Hamiltonian and temperature, the Davies master equation describes a Markovian open system that converges to the Gibbs state of the Hamiltonian at that temperature. While the quantum Metropolis algorithm [40,47] can be used to prepare this Gibbs state, simulating the Davies equation would provide a method of simulating the approach to equilibrium. Note that the overcomplete GKS matrix for the Davies master equation is diagonal in the eigenbasis of its system Hamiltonian, but it is not obvious how to apply our methods in that basis.…”