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Volume 11 (2015) Article 20 pp. 491-603
The Bose-Hubbard Model is QMA-complete
Received: July 31, 2014
Revised: June 25, 2015
Published: December 31, 2015
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Keywords: complexity theory, QMA, quantum computing
ACM Classification: F.1.3,F.2.2
AMS Classification: 68Q17, 81P68

Abstract: [Plain Text Version]

The Bose-Hubbard model is a system of interacting bosons that live on the vertices of a graph. The particles can move between adjacent vertices and experience a repulsive on-site interaction. The Hamiltonian is determined by a choice of graph that specifies the geometry in which the particles move and interact. We prove that approximating the ground energy of the Bose-Hubbard model on a graph at fixed particle number is QMA-complete. In our QMA-hardness proof, we encode the history of an $n$-qubit computation in the subspace with at most one particle per site (i.e., hard-core bosons). This feature, along with the well-known mapping between hard-core bosons and spin systems, lets us prove a related result for a class of $2$-local Hamiltonians defined by graphs that generalizes the XY model. By avoiding the use of perturbation theory in our analysis, we circumvent the need to multiply terms in the Hamiltonian by large coefficients.