A Quantum-Quantum Metropolis Algorithm

TL;DR

This paper introduces a quantum Metropolis algorithm that generalizes classical Metropolis sampling to quantum Hamiltonians, achieving a quadratic quantum speedup in eigenvalue gap for any quantum Hamiltonian.

Contribution

It presents a novel quantum Metropolis algorithm that extends classical methods to the quantum domain with proven quadratic speedup.

Findings

Achieves quadratic quantum speedup in eigenvalue gap

Generalizes classical Metropolis to quantum Hamiltonians

Applicable to any quantum Hamiltonian

Abstract

Recently, the idea of classical Metropolis sampling through Markov chains has been generalized for quantum Hamiltonians. However, the underlying Markov chain of this algorithm is still classical in nature. Due to Szegedy's method, the Markov chains of classical Hamiltonians can achieve a quadratic quantum speedup in the eigenvalue gap of the corresponding transition matrix. A natural question to ask is whether Szegedy's quantum speedup is merely a consequence of employing classical Hamiltonians, where the eigenstates simply coincide with the computational basis, making cloning of the classical information possible. We solve this problem by introducing a quantum version of the method of Markov-chain quantization combined with the quantum simulated annealing (QSA) procedure, and describe explicitly a novel quantum Metropolis algorithm, which exhibits a quadratic quantum speedup in the eigenvalue gap of the corresponding Metropolis Markov chain for any quantum Hamiltonian. This result provides a complete generalization of the classical Metropolis method to the quantum domain.