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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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