Markov chain Monte Carlo enhanced variational quantum algorithms

Abstract

Variational quantum algorithms have the potential for significant impact on high-dimensional optimization, with applications in classical combinatorics, quantum chemistry, and condensed matter. Nevertheless, the optimization landscape of these algorithms is generally nonconvex, leading the algorithms to converge to local, rather than global, minima and the production of suboptimal solutions. In this work, we introduce a variational quantum algorithm that couples classical Markov chain Monte Carlo techniques with variational quantum algorithms, allowing the former to provably converge to global minima and thus assure solution quality. Due to the generality of our approach, it is suitable for a myriad of quantum minimization problems, including optimization and quantum state preparation. Specifically, we devise a Metropolis-Hastings method that is suitable for variational quantum devices and use it, in conjunction with quantum optimization, to construct quantum ensembles that converge to Gibbs states. These performance guarantees are derived from the ergodicity of our algorithm's state space and enable us to place analytic bounds on its time-complexity. We demonstrate both the effectiveness of our technique and the validity of our analysis through quantum circuit simulations for MaxCut instances, solving these problems deterministically and with perfect accuracy, as well as large-scale quantum Ising and transverse field spin models of up to $50$ qubits. Our technique stands to broadly enrich the field of variational quantum algorithms, improving and guaranteeing the performance of these promising, yet often heuristic, methods.

SIC Error Towards the Packet Error Rate: Optimal Partitioning in OTFS-NOMA Systems

“…REMARK 4. If a Markov chain is: (i) irreducible 13 ; and (ii) aperiodic, it has a unique stationary distribution [40], [41], [42], [43].…”

“…REMARK 5. If our Markov chain has a unique stationary distribution, one can prove the optimality of the cost/utility function as well [40], [41], [42], [43]. Indeed…”

“…DEFINITION 5: MARKOV CHAIN MIXING TIME [40], [41], [42], [43]. Markov chain mixing time is the time interval that should be over in the sense that the Markov chain meets its steady state distribution − something that is upperbounded to the Kullback-Leibler divergence between the given distribution of the Markov chain and the true unique stationary distribution, i.e., KL 𝜋||𝜋 (𝓊) where 𝜋 (𝓊) denotes the later one.…”

Ising Hamiltonians for Constrained Combinatorial Optimization Problems and the Metropolis‐Hastings Warm‐Starting Algorithm

“…While low-depth QAOA may achieve promising results for some problems, [7] the optimization landscapes would become more nonconvex so that it tends to obtain the local optimal solutions which plagues QAOA. [16,17] Recently, the warm-starting algorithms [17][18][19] have been proposed to tackle this issue. Especially, the classical Metropolis-Hastings algorithm has been utilized as a warm-start for VQE to avoid the local minima convergence, due to its provable ergodicity and suitability for unnormalized probability distributions.…”

“…Especially, the classical Metropolis-Hastings algorithm has been utilized as a warm-start for VQE to avoid the local minima convergence, due to its provable ergodicity and suitability for unnormalized probability distributions. [17] Moreover, for an ergodic, discrete-time Markov chain, the number of epochs required to reach a certain threshold of convergence is analytically bounded by…”