Quantum Algorithms for Simulated Annealing

Abstract

This problem is concerned with the development of quantum methods to speed up classical algorithms based on simulated annealing (SA).SA is a well known and powerful strategy to solve discrete combinatorial optimization problems [1]. The search space Σ = {σ 0 , . . . , σ d−1 } consists of d configurations σ i and the goal is to find the (optimal) configuration that corresponds to the global minimum of a given cost function E : Σ → R. Monte Carlo implementations of SA generate a stochastic sequence of configurations via a sequence of Markov processes that converges to the lowtemperature Gibbs (probability) distribution, π βm (Σ) ∝ exp(−β m E(Σ)). If β m is sufficiently large, sampling from the Gibbs distribution outputs an optimal configuration with large probability, thus solving the combinatorial optimization problem. The annealing process depends on the choice of an annealing schedule, which consists of a sequence of d × d stochastic matrices (transition rules) S(β 1 ), S(β 2 ), . . . , S(β m ). Such matrices are determined, e.g., by using Metropolis-Hastings [2]. The real parameters β j denote a sequence of "inverse temperatures". The implementation complexity of SA is given by m, the number of times that transition rules must be applied to converge to the desired Gibbs distribution (within arbitrary precision). Commonly, the stochastic matrices are sparse and each list of nonzero conditional probabilities and corresponding configurations, {Pr β (σ j |σ i ), j : Pr β (σ j |σ i ) > 0}, can be efficiently computed on input (i, β). This implies an efficient Monte Carlo implementation of each Markov process. When a lower bound on the spectral gap of the stochastic matrices (i.e., the difference between the two largest eigenvalues) is known and given by ∆ > 0, one can choose (β k+1 − β k ) ∝ ∆/E max and β 0 = 0, β m ∝ log √ d.

Hamiltonian simulation in the low-energy subspace

“…To this end, we investigate the Hamiltonian simulation problem when the initial state is supported on a low-energy subspace. This is a central problem in physics that has vast applications, including the simulation of condensed matter systems for studying quantum phase transitions 25 , the simulation of quantum field theories 13 , the simulation of adiabatic quantum state preparation 26,27 , and more. We analyze the complexities of product formulas in this setting and show significant improvements with respect to the best-known complexity bounds that apply to the general case.…”