[Paper Review] Monte Carlo simulation of stoquastic Hamiltonians

Sergey Bravyi|arXiv (Cornell University)|Feb 10, 2014

Quantum many-body systems27 references12 citations

TL;DR

This paper establishes that the ground state energy of local stoquastic Hamiltonians can be efficiently estimated using Monte Carlo methods when a guiding state with non-negligible overlap to the ground state exists. It proves the Guided Stoquastic Hamiltonian problem is complete for the complexity class MA and shows that the ferromagnetic Transverse-Field Ising Model can be simulated in polynomial time via a classical FPRAS based on the Jerrum-Sinclair algorithm.

ABSTRACT

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).

Motivation & Objective

• To formalize the computational complexity of estimating ground state energies for stoquastic Hamiltonians with a guiding state.
• To show that the Guided Stoquastic Hamiltonian problem is complete for the complexity class MA.
• To demonstrate that the ground state and thermal properties of the ferromagnetic Transverse-Field Ising Model can be classically simulated in polynomial time.
• To provide a rigorous foundation for the belief that stoquastic Hamiltonians avoid the sign problem and are amenable to efficient classical simulation.

Proposed method

• Uses the Projection Monte Carlo algorithm with a variable number of walkers to simulate stoquastic Hamiltonians.
• Applies the Suzuki-Trotter formula to map the quantum partition function to a classical partition function with non-negative weights.
• Employs a guiding state with polynomially small but non-negligible overlap to the ground state to enable efficient sampling.
• Reduces the problem to a classical ferromagnetic Ising model by mapping multiple copies of the system with inter-layer couplings.
• Utilizes the FPRAS (Fully Polynomial Randomized Approximation Scheme) for the classical ferromagnetic Ising model due to Jerrum and Sinclair (1993).
• Implements error analysis using Taylor series expansions and operator norm bounds to control approximation errors in the Trotter-Suzuki decomposition.

Experimental results

Research questions

• RQ1 Is the problem of estimating the ground state energy of a stoquastic Hamiltonian with a guiding state contained in the complexity class MA?
• RQ2 Can the ground state energy of the ferromagnetic Transverse-Field Ising Model be efficiently simulated on a classical computer?
• RQ3 Does the existence of a guiding state with non-negligible overlap enable efficient classical Monte Carlo simulation of stoquastic Hamiltonians?
• RQ4 Can the sign problem be avoided in a broad class of quantum models through a basis transformation and classical sampling?
• RQ5 What is the computational complexity of the Guided Stoquastic Hamiltonian problem?

Key findings

• The Guided Stoquastic Hamiltonian problem is complete for the complexity class MA, establishing its computational hardness and verifiability under probabilistic proof.
• The ground state energy of the ferromagnetic Transverse-Field Ising Model can be approximated within multiplicative error δ in time O(n^59 J^21 δ^(-9)), ignoring logarithmic factors.
• The classical partition function of the mapped ferromagnetic Ising model admits an FPRAS with running time O(δ^(-2) M^3 N^11 log N), where N = nr and M ≤ n^2 r.
• The approximation error in the quantum partition function is bounded by δ through a perturbation argument assuming h_u ≥ δ/n.
• The operator norm of the error in the Trotter-Suzuki approximation is bounded by 12ρ^3, ensuring convergence of the series expansion.
• The method achieves a polynomial-time classical simulation of the thermal and ground state properties of the ferromagnetic TIM, demonstrating that such models are efficiently simulable despite quantum origin.