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Quantum Information and Computation     ISSN: 1533-7146      published since 2001
Vol.15 No.13&14  October 2015

Monte Carlo simulation of stoquastic Hamiltonians (pp1122-1140)
          
Sergey Bravyi
         
doi: https://doi.org/10.26421/QIC15.13-14-3

Abstracts: Stoquastic Hamiltonians are characterized by the property that their off-diagonal matrix elements in the standard product basis are real and non-positive. Many interesting quantum models fall into this class including the Transverse field Ising Model (TIM), the Heisenberg model on bipartite graphs, and the bosonic Hubbard model. Here we consider the problem of estimating the ground state energy of a local stoquastic Hamiltonian H with a promise that the ground state of H has a non-negligible correlation with some ‘guiding’ state that admits a concise classical description. A formalized version of this problem called Guided Stoquastic Hamiltonian is shown to be complete for the complexity class MA (a probabilistic analogue of NP). To prove this result we employ the Projection Monte Carlo algorithm with a variable number of walkers. Secondly, we show that the ground state and thermal equilibrium properties of the ferromagnetic TIM can be simulated in polynomial time on a classical probabilistic computer. This result is based on the approximation algorithm for the classical ferromagnetic Ising model due to Jerrum and Sinclair (1993).
Key words: Quantum Monte Carlo, fermionic sign problem, transverse Ising model

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