  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.
