Computational Difficulty of Computing the Density of States

TL;DR

This paper investigates the computational complexity of calculating the density of states and ground state degeneracy for local Hamiltonians, revealing their equivalence in difficulty for classical and quantum cases through the #BQP complexity class.

Contribution

It introduces the #BQP class to precisely characterize the complexity of these problems and shows their computational hardness is comparable for classical and quantum Hamiltonians.

Findings

Computing density of states is #BQP-complete.

Ground state degeneracy computation is as hard for classical as for quantum Hamiltonians.

#BQP is not harder than #P, linking quantum and classical complexity.

Abstract

We study the computational difficulty of computing the ground state degeneracy and the density of states for local Hamiltonians. We show that the difficulty of both problems is exactly captured by a class which we call #BQP, which is the counting version of the quantum complexity class QMA. We show that #BQP is not harder than its classical counting counterpart #P, which in turn implies that computing the ground state degeneracy or the density of states for classical Hamiltonians is just as hard as it is for quantum Hamiltonians.