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1. Rapid mixing of path integral Monte Carlo for 1D stoquastic Hamiltonians

   https://doi.org/10.22331/q-2021-02-11-395  

   Crosson, Elizabeth; Harrow, Aram W. (February 2021, Quantum)

   null (Ed.)

   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. 

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2. Unbiasing fermionic quantum Monte Carlo with a quantum computer

   https://doi.org/10.1038/s41586-021-04351-z  

   Huggins, William J.; O’Gorman, Bryan A.; Rubin, Nicholas C.; Reichman, David R.; Babbush, Ryan; Lee, Joonho (March 2022, Nature)

   Abstract Interacting many-electron problems pose some of the greatest computational challenges in science, with essential applications across many fields. The solutions to these problems will offer accurate predictions of chemical reactivity and kinetics, and other properties of quantum systems 1–4 . Fermionic quantum Monte Carlo (QMC) methods 5,6 , which use a statistical sampling of the ground state, are among the most powerful approaches to these problems. Controlling the fermionic sign problem with constraints ensures the efficiency of QMC at the expense of potentially significant biases owing to the limited flexibility of classical computation. Here we propose an approach that combines constrained QMC with quantum computation to reduce such biases. We implement our scheme experimentally using up to 16 qubits to unbias constrained QMC calculations performed on chemical systems with as many as 120 orbitals. These experiments represent the largest chemistry simulations performed with the help of quantum computers, while achieving accuracy that is competitive with state-of-the-art classical methods without burdensome error mitigation. Compared with the popular variational quantum eigensolver 7,8 , our hybrid quantum-classical computational model offers an alternative path towards achieving a practical quantum advantage for the electronic structure problem without demanding exceedingly accurate preparation and measurement of the ground-state wavefunction. 

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3. Steady-state properties of multi-orbital systems using quantum Monte Carlo

   https://doi.org/10.1063/5.0226253  

   Erpenbeck, A; Blommel, T; Zhang, L; Lin, W-T; Cohen, G; Gull, E (September 2024, The Journal of Chemical Physics)

   A precise dynamical characterization of quantum impurity models with multiple interacting orbitals is challenging. In quantum Monte Carlo methods, this is embodied by sign problems. A dynamical sign problem makes it exponentially difficult to simulate long times. A multi-orbital sign problem generally results in a prohibitive computational cost for systems with multiple impurity degrees of freedom even in static equilibrium calculations. Here, we present a numerically exact inchworm method that simultaneously alleviates both sign problems, enabling simulation of multi-orbital systems directly in the equilibrium or nonequilibrium steady-state. The method combines ideas from the recently developed steady-state inchworm Monte Carlo framework [Erpenbeck et al., Phys. Rev. Lett. 130, 186301 (2023)] with other ideas from the equilibrium multi-orbital inchworm algorithm [Eidelstein et al., Phys. Rev. Lett. 124, 206405 (2020)]. We verify our method by comparison with analytical limits and numerical results from previous methods. 

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4. Monte Carlo methods in the manifold of Hartree–Fock–Bogoliubov wave functions

   https://doi.org/10.1063/5.0228450  

   Vitali, Ettore; Rosenberg, Peter; Zhang, Shiwei (October 2024, The Journal of Chemical Physics)

   We explore the possibility of implementing random walks in the manifold of Hartree–Fock–Bogoliubov wave functions. The goal is to extend state-of-the-art quantum Monte Carlo approaches, in particular the constrained-path auxiliary-field quantum Monte Carlo technique, to systems where finite pairing order parameters or complex pairing mechanisms, e.g., Fulde–Ferrell–Larkin–Ovchinnikov pairing or triplet pairing, may be expected. Leveraging the flexibility to define a vacuum state tailored to the physical problem, we discuss a method to use imaginary-time evolution of Hartree–Fock–Bogoliubov states to compute ground state correlations, extending beyond situations spanned by current formalisms. Illustrative examples are provided. 

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5. Numerical analysis for inchworm Monte Carlo method: Sign problem and error growth

   https://doi.org/10.1090/mcom/3785  

   Cai, Zhenning; Lu, Jianfeng; Yang, Siyao (May 2023, Mathematics of Computation)

   We consider the numerical analysis of the inchworm Monte Carlo method, which is proposed recently to tackle the numerical sign problem for open quantum systems. We focus on the growth of the numerical error with respect to the simulation time, for which the inchworm Monte Carlo method shows a flatter curve than the direct application of Monte Carlo method to the classical Dyson series. To better understand the underlying mechanism of the inchworm Monte Carlo method, we distinguish two types of exponential error growth, which are known as the numerical sign problem and the error amplification. The former is due to the fast growth of variance in the stochastic method, which can be observed from the Dyson series, and the latter comes from the evolution of the numerical solution. Our analysis demonstrates that the technique of partial resummation can be considered as a tool to balance these two types of error, and the inchworm Monte Carlo method is a successful case where the numerical sign problem is effectively suppressed by such means. We first demonstrate our idea in the context of ordinary differential equations, and then provide complete analysis for the inchworm Monte Carlo method. Several numerical experiments are carried out to verify our theoretical results. 

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