The Complexity of the Separable Hamiltonian Problem

TL;DR

This paper investigates the complexity of quantum Hamiltonian problems with separability constraints, showing that the Separable-Sparse-Hamiltonian problem is QMA(2)-Complete, highlighting a new complexity class distinction.

Contribution

It introduces and analyzes the Separable-Sparse-Hamiltonian problem, proving it is QMA(2)-Complete, unlike the standard Local-Hamiltonian problem which is QMA.

Findings

Separable-Sparse-Hamiltonian is QMA(2)-Complete.

Separable-Local-Hamiltonian is in QMA.

First non-trivial problem shown to be QMA(2)-Complete.

Abstract

In this paper, we study variants of the canonical Local-Hamiltonian problem where, in addition, the witness is promised to be separable. We define two variants of the Local-Hamiltonian problem. The input for the Separable-Local-Hamiltonian problem is the same as the Local-Hamiltonian problem, i.e. a local Hamiltonian and two energies a and b, but the question is somewhat different: the answer is YES if there is a separable quantum state with energy at most a, and the answer is NO if all separable quantum states have energy at least b. The Separable-Sparse-Hamiltonian problem is defined similarly, but the Hamiltonian is not necessarily local, but rather sparse. We show that the Separable-Sparse-Hamiltonian problem is QMA(2)-Complete, while Separable-Local-Hamiltonian is in QMA. This should be compared to the Local-Hamiltonian problem, and the Sparse-Hamiltonian problem which are both QMA-Complete. To the best of our knowledge, Separable-SPARSE-Hamiltonian is the first non-trivial problem shown to be QMA(2)-Complete.