Linear time algorithm for quantum 2SAT

TL;DR

This paper presents a linear-time classical algorithm for solving the quantum 2-SAT problem, extending classical satisfiability results to the quantum domain with optimal efficiency.

Contribution

The authors develop the first linear-time algorithm for quantum 2-SAT, improving upon previous polynomial-time solutions and matching the problem's input size.

Findings

The algorithm runs in linear time relative to the number of projectors.

It solves quantum 2-SAT efficiently, matching the lower bound of input size.

The method advances understanding of quantum satisfiability and Hamiltonian ground states.

Abstract

A canonical result about satisfiability theory is that the 2-SAT problem can be solved in linear time, despite the NP-hardness of the 3-SAT problem. In the quantum 2-SAT problem, we are given a family of 2-qubit projectors $\Pi_{ij}$ on a system of $n$ qubits, and the task is to decide whether the Hamiltonian $H=\sum \Pi_{ij}$ has a 0-eigenvalue, or it is larger than $1/n^\alpha$ for some $\alpha=O(1)$. The problem is not only a natural extension of the classical 2-SAT problem to the quantum case, but is also equivalent to the problem of finding the ground state of 2-local frustration-free Hamiltonians of spin $\frac{1}{2}$, a well-studied model believed to capture certain key properties in modern condensed matter physics. While Bravyi has shown that the quantum 2-SAT problem has a classical polynomial-time algorithm, the running time of his algorithm is $O(n^4)$. In this paper we give a classical algorithm with linear running time in the number of local projectors, therefore achieving the best possible complexity.