# Hardness and Ease of Curing the Sign Problem for Two-Local Qubit Hamiltonians

## TL;DR

This paper investigates the computational complexity of transforming two-local qubit Hamiltonians into a form without the sign problem, revealing NP-hardness with one-local terms and polynomial-time solvability without them.

## Contribution

It establishes the NP-hardness of curing the sign problem with one-local terms and provides a polynomial-time algorithm for cases without such terms.

## Key findings

- NP-hardness when one-local terms are present
- Polynomial-time algorithm for Hamiltonians without one-local terms
- Deciding the sign problem's cure is computationally complex in general

## Abstract

We examine the problem of determining whether a multi-qubit two-local Hamiltonian can be made stoquastic by single-qubit unitary transformations. We prove that when such a Hamiltonian contains one-local terms, then this task can be NP-hard. This is shown by constructing a class of Hamiltonians for which performing this task is equivalent to deciding $3$-SAT. In contrast, we show that when such a Hamiltonian contains no one-local terms then this task is easy, namely we present an algorithm which decides, in a number of arithmetic operations over $\mathbb{R}$ which is polynomial in the number of qubits, whether the sign problem of the Hamiltonian can be cured by single-qubit rotations.