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arXiv · 1203.3906

The k-local Pauli Commuting Hamiltonians Problem is in P

Jijiang Yan·Dave Bacon
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Abstract

Given a Hamiltonian that is a sum of commuting few-body terms, the commuting Hamiltonian problem is to determine if there exists a quantum state that is the simultaneous eigenstate of all of these terms that minimizes each term individually. This problem is known to be in the complexity class quantum Merlin-Arthur, but is widely thought to not be complete for this class. Here we show that a limited form of this problem when the individual terms are all made up of tensor products of Pauli matrices is efficiently solvable on a classical computer and thus in the complexity class P. The problem can be thought of as the classical XOR-SAT problem over a symplectic vector space. This class of problems includes instance Hamiltonians whose ground states possess topological entanglement, thus showing that such entanglement is not always a barrier for the more general problem.

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Jijiang Yan, Dave Bacon. 2012-03-17. The k-local Pauli Commuting Hamiltonians Problem is in P. https://arxiv.org/abs/1203.3906

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