On the Computational Complexity of Curing the Sign Problem

TL;DR

This paper investigates the complexity of transforming non-stoquastic Hamiltonians into sign-problem-free forms, revealing that such curing transformations are NP-complete under certain restrictions, highlighting fundamental computational limitations.

Contribution

It proves that finding curing transformations limited to single-qubit Clifford or orthogonal matrices is NP-complete, establishing a complexity barrier in quantum simulation.

Findings

Curing transformations are NP-complete under specified restrictions.

Implications for the efficiency of quantum Monte Carlo algorithms.

Highlights fundamental computational challenges in quantum Hamiltonian transformations.

Abstract

Quantum many-body systems whose Hamiltonians are non-stoquastic, i.e., have positive off-diagonal matrix elements in a given basis, are known to pose severe limitations on the efficiency of Quantum Monte Carlo algorithms designed to simulate them, due to the infamous sign problem. We study the computational complexity associated with `curing' non-stoquastic Hamiltonians, i.e., transforming them into sign-problem-free ones. We prove that if such transformations are limited to single-qubit Clifford group elements or general single-qubit orthogonal matrices, finding the curing transformation is NP-complete. We discuss the implications of this result.