Multi-step quantum algorithm for solving the 3-bit exact cover problem

TL;DR

This paper introduces a multi-step quantum algorithm that efficiently solves the 3-bit exact cover problem, an NP-complete problem, by sequentially applying clauses and reducing the search space exponentially.

Contribution

The paper presents a novel quantum algorithm that reduces the search space exponentially and scales polynomially, outperforming brute force methods for NP-complete problems.

Findings

Search space reduced exponentially

Runtime scales polynomially with problem size

Quantum approach may outperform classical methods

Abstract

We present a multi-step quantum algorithm for solving the $3$-bit exact cover problem, which is one of the NP-complete problems. Unlike the brute force methods have been tried before, in this algorithm, we showed that by applying the clauses of the Boolean formula sequentially and introducing non-unitary operations, the state that satisfies all of the clauses can be projected out from an equal superposition of all computational basis states step by step, and the search space is reduced exponentially. The runtime of the algorithm is proportional to the number of clauses, therefore scales polynomial to the size of the problem. Our results indicate that quantum computers may be able to outperform classical computers in solving NP-complete problems.