Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations

TL;DR

This paper proves that solving the fermionic quantum Monte Carlo sign problem efficiently is almost certainly impossible by demonstrating its NP-hardness, indicating fundamental computational limitations.

Contribution

It establishes the NP-hardness of the fermionic sign problem, providing a theoretical barrier to polynomial-time solutions for this quantum simulation challenge.

Findings

Sign problem is NP-hard for fermionic quantum Monte Carlo

No polynomial-time solution likely exists for the sign problem

Fundamental computational limitations are identified for fermionic simulations

Abstract

Quantum Monte Carlo simulations, while being efficient for bosons, suffer from the "negative sign problem'' when applied to fermions - causing an exponential increase of the computing time with the number of particles. A polynomial time solution to the sign problem is highly desired since it would provide an unbiased and numerically exact method to simulate correlated quantum systems. Here we show, that such a solution is almost certainly unattainable by proving that the sign problem is NP-hard, implying that a generic solution of the sign problem would also solve all problems in the complexity class NP (nondeterministic polynomial) in polynomial time.