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The Bose-Hubbard Model is QMA-complete

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Automata, Languages, and Programming (ICALP 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8572))

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Abstract

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.

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Author information

Authors and Affiliations

  1. Department of Combinatorics & Optimization, University of Waterloo, Canada

    Andrew M. Childs & David Gosset

  2. Institute for Quantum Computing, University of Waterloo, Canada

    Andrew M. Childs, David Gosset & Zak Webb

  3. Department of Physics & Astronomy, University of Waterloo, Canada

    Zak Webb

Authors
  1. Andrew M. Childs
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  2. David Gosset
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  3. Zak Webb
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Editor information

Editors and Affiliations

  1. Institut für Informatik, Technische Universität München, Boltzmannstrasse 3, 85748, München, Germany

    Javier Esparza

  2. LIAFA, Université Paris Diderot-Paris 7, Case 7014, 75205, Paris Cedex 13, France

    Pierre Fraigniaud

  3. IT University of Copenhagen, Rued Langgaards Vej 7, 2300, Copenhagen, Denmark

    Thore Husfeldt

  4. Department Computer Science, University of Oxford, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK

    Elias Koutsoupias

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Childs, A.M., Gosset, D., Webb, Z. (2014). The Bose-Hubbard Model is QMA-complete. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds) Automata, Languages, and Programming. ICALP 2014. Lecture Notes in Computer Science, vol 8572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43948-7_26

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  • DOI: https://doi.org/10.1007/978-3-662-43948-7_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-43947-0

  • Online ISBN: 978-3-662-43948-7

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