Interacting boson problems are QMA-hard

TL;DR

This paper demonstrates that the computational complexity of interacting boson problems is as hard as fermionic problems, establishing their QMA-hardness and QMA-completeness, which implies no efficient quantum algorithms are likely for these problems.

Contribution

It proves that interacting boson problems and the bosonic N-representability problem are QMA-hard and QMA-complete, challenging the conventional view of their tractability.

Findings

Interacting boson problems are QMA-hard.

Bosonic N-representability problem is QMA-complete.

These problems are unlikely to have efficient quantum algorithms.

Abstract

Computing the ground-state energy of interacting electron (fermion) problems has recently been shown to be hard for QMA, a quantum analogue of the complexity class NP. Fermionic problems are usually hard, a phenomenon widely attributed to the so-called sign problem occurring in Quantum Monte Carlo simulations. The corresponding bosonic problems are, according to conventional wisdom, tractable. Here, we discuss the complexity of interacting boson problems and show that they are also QMA-hard. In addition, we show that the bosonic version of the so-called N-representability problem is QMA-complete, as hard as its fermionic version. As a consequence, these problems are unlikely to have efficient quantum algorithms.