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.
Highlights
The problem of approximating the ground energy of a given Hamiltonian is a natural quantum analog of classical constraint satisfaction
For a variety of classes of Hamiltonians and a suitable notion of approximation, this task is complete for the complexity class QMA, the quantum version of NP with two-sided error. These results provide evidence that approximating the ground energy of such quantum systems is intractable
The first such example is the Local Hamiltonian problem introduced by Kitaev [21]
Summary
The problem of approximating the ground energy of a given Hamiltonian is a natural quantum analog of classical constraint satisfaction. The QMA-hardness of ground energy problems for local Hamiltonians acting on qubits has implications for Hamiltonians acting on indistinguishable particles (bosons or fermions) due to formal mappings between these systems. A more restrictive class of QMA-complete fermionic Hamiltonians was considered by Schuch and Verstraete, who showed that the Hubbard model with a site-dependent magnetic field is QMA-complete [29]. This is a specific model of interacting electrons The dynamics of stoquastic local Hamiltonians are BQP-complete (as follows from [16] and time reversal), whereas the corresponding ground energy problem is in AM [5] and unlikely to be QMA-hard. The ferromagnetic Heisenberg model on a graph provides an even starker contrast: its dynamics are BQPcomplete (as can be inferred from [8] using a correspondence between spins and hard-core bosons) but its ground energy problem is trivial since the ground space is the symmetric subspace
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