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Monte Carlo simulation of stoquastic Hamiltonians
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Monte Carlo simulation of stoquastic Hamiltonians
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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 Jerrrum and Sinclair (1993).
Forward citations
Cited by 7 Pith papers
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The Guided Local Hamiltonian Problem for Stoquastic Hamiltonians
The Guided Local Hamiltonian problem for stoquastic Hamiltonians is promise BPP-hard (even 2-local on lattices), BQP-hard under fixed local constraints, and admits a deterministic classical approximation algorithm whe...
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RFOX keeps the instantaneous spectral gap flat across interpolation and disorder by using a constant XX catalyst plus derived ZX counter-diabatic drive, yielding faster ground-state convergence on small RFIM instances.
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RFOX (Rotated-Field Oscillatory eXchange) quantum algorithm: Towards Parameter-Free Quantum Optimizers
RFOX maintains a flat spectral gap via non-stoquastic XX catalyst plus analytic counter-diabatic ZX driving, yielding near-optimal solutions on random-field Ising models with up to 10x fewer Trotter steps.
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On the Complexity of the Succinct State Local Hamiltonian Problem
The succinct state 2-local Hamiltonian problem for qubit Hamiltonians is promise-MA-complete.
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The Complexity of Local Stoquastic Hamiltonians on 2D Lattices
The 2-local stoquastic Hamiltonian problem on 2D square qubit lattices is StoqMA-complete.
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