A Quantum Version of Sch\"oning's Algorithm Applied to Quantum 2-SAT

TL;DR

This paper introduces a quantum analogue of Sch"oning's algorithm for Quantum 2-SAT, demonstrating its efficiency in solving the problem and producing high-overlap states under certain conditions.

Contribution

It presents a quantum Markov process algorithm for Quantum 2-SAT, analyzing its performance and efficiency, extending classical probabilistic methods into the quantum domain.

Findings

Algorithm solves Quantum 2-SAT with high probability in polynomial time

Efficiently produces states with high overlap with the satisfying subspace

Performance depends on the promise gap and Hamiltonian gap

Abstract

We study a quantum algorithm that consists of a simple quantum Markov process, and we analyze its behavior on restricted versions of Quantum 2-SAT. We prove that the algorithm solves this decision problem with high probability for n qubits, L clauses, and promise gap c in time O(n^2 L^2 c^{-2}). If the Hamiltonian is additionally polynomially gapped, our algorithm efficiently produces a state that has high overlap with the satisfying subspace. The Markov process we study is a quantum analogue of Sch\"oning's probabilistic algorithm for k-SAT.