Abstract
Path integral quantum Monte Carlo (PIMC) is a method for estimating thermal equilibrium properties of stoquastic quantum spin systems by sampling from a classical Gibbs distribution using Markov chain Monte Carlo. The PIMC method has been widely used to study the physics of materials and for simulated quantum annealing, but these successful applications are rarely accompanied by formal proofs that the Markov chains underlying PIMC rapidly converge to the desired equilibrium distribution. In this work we analyze the mixing time of PIMC for 1D stoquastic Hamiltonians, including disordered transverse Ising models (TIM) with long-range algebraically decaying interactions as well as disordered XY spin chains with nearest-neighbor interactions. By bounding the convergence time to the equilibrium distribution we rigorously justify the use of PIMC to approximate partition functions and expectations of observables for these models at inverse temperatures that scale at most logarithmically with the number of qubits. The mixing time analysis is based on the canonical paths method applied to the single-site Metropolis Markov chain for the Gibbs distribution of 2D classical spin models with couplings related to the interactions in the quantum Hamiltonian. Since the system has strongly nonisotropic couplings that grow with system size, it does not fall into the known cases where 2D classical spin models are known to mix rapidly.
Highlights
The application of Markov chain Monte Carlo methods to thermal states of a limited class of quantum systems was first proposed by Suzuki, Miyashita and Kuroda 40 years ago [43]
Many quantum systems of physical and computational interest are stoquastic, including spinless particles moving on arbitrary graphs with position dependent interactions, as well as generalized transverse Ising models, which are notable for their use in quantum annealing [1, 19, 31]
We show the mixing time of PIMC is at most poly(n, eβ, Γ−1) for generalized transverse Ising models (TIM) with σxσx and σyσy interactions, n n n n
Summary
The application of Markov chain Monte Carlo methods to thermal states of a limited class of quantum systems was first proposed by Suzuki, Miyashita and Kuroda 40 years ago [43]. These methods do not require the Hamiltonian to be stoquastic and have been applied in rigorous classical simulations of systems with limited entanglement [4, 22, 33] Their runtime scales exponentially with spectral gap, while our runtime bound for PIMC scales exponentially with the inverse temperature, which is qualitatively similar.
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