The Complexity of the Consistency and N-representability Problems for Quantum States

TL;DR

This paper establishes the QMA-completeness of several problems related to quantum states, including local density matrix consistency, fermionic Hamiltonians, and N-representability, advancing understanding of their computational complexity.

Contribution

It proves the QMA-completeness of the consistency of local density matrices, fermionic local Hamiltonian, and N-representability problems, linking quantum chemistry and quantum complexity theory.

Findings

Consistency of local density matrices is QMA-complete.

Fermionic local Hamiltonian problem is QMA-complete.

N-representability problem is QMA-complete.

Abstract

QMA (Quantum Merlin-Arthur) is the quantum analogue of the class NP. There are a few QMA-complete problems, most notably the ``Local Hamiltonian'' problem introduced by Kitaev. In this dissertation we show some new QMA-complete problems. The first one is ``Consistency of Local Density Matrices'': given several density matrices describing different (constant-size) subsets of an n-qubit system, decide whether these are consistent with a single global state. This problem was first suggested by Aharonov. We show that it is QMA-complete, via an oracle reduction from Local Hamiltonian. This uses algorithms for convex optimization with a membership oracle, due to Yudin and Nemirovskii. Next we show that two problems from quantum chemistry, ``Fermionic Local Hamiltonian'' and ``N-representability,'' are QMA-complete. These problems arise in calculating the ground state energies of molecular systems. N-representability is a key component in recently developed numerical methods using the contracted Schrodinger equation. Although these problems have been studied since the 1960's, it is only recently that the theory of quantum computation has allowed us to properly characterize their complexity. Finally, we study some special cases of the Consistency problem, pertaining to 1-dimensional and ``stoquastic'' systems. We also give an alternative proof of a result due to Jaynes: whenever local density matrices are consistent, they are consistent with a Gibbs state.