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

TitleThe Complexity of the Consistency and N-representability Problems for Quantum States
Publication TypeJournal Article
Year of Publication2007
AuthorsLiu, Y-K
Date Published2007/12/18
Type of ArticlePhD Thesis, Univ. of California, San Diego

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
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.