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Brazilian Journal of Physics

Histogram methods for quantum systems: from reweighting to Wang-Landau sampling

Matthias Troyer, Fabien Alet, Stefan Weßel

Published Jun 1, 2004• 26 citations
Publisher

Abstract

We discuss how reweighting and histogram methods for classical systems can be generalized to quantum models for discrete and continuous time world line simulations, and the stochastic series expansion (SSE) method. Our approach allows to apply all classical reweighting and histogram techniques for classical systems, as well as multicanonical or Wang-Landau sampling to the quantum case.

DOI
10.1590/s0103-97332004000300008

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We discuss how reweighting and histogram methods for classical systems can be generalized to quantum models for discrete and continuous time world line simulations, and the stochastic series expansion (SSE) method. Our approach allows to apply all classical reweighting and histogram techniques for classical systems, as well as multicanonical or Wang-Landau sampling to the quantum case.
The Stochastic Series Expansion (SSE) technique is a quantum Monte Carlo method that is especially efficient for many quantum spin systems and boson models. It was the first generic method free from the discretization errors affecting previous path integral based approaches. These lecture notes give a brief overview of the SSE method and its applications. In the introductory section, the representation of quantum statistical mechanics by the power series expansion of e−βH{\rm e}^{-βH}e−βH will be compared with path integrals in discrete and continuous imaginary time. Extensions of the SSE approach to ground state projection and quantum annealing in imaginary time will also be briefly discussed. The later sections introduce efficient sampling schemes (loop and cluster updates) that have been developed for many classes of models. A summary of generic forms of estimators for important observables are also given. Applications are discussed in the last section.
The efficiency of statistical sampling in broad-histogram Monte Carlo simulations can be considerably improved by optimizing the simulated extended ensemble for fastest equilibration. Here we describe how a recently developed feedback algorithm can be generalized to find optimized sampling distributions for the simulation of quantum systems in the context of the stochastic series expansion (SSE) when defining an extended ensemble in the expansion order. If the chosen update method is efficient, such as non-local updates for systems undergoing a second-order phase transition, the optimized ensemble is characterized by a flat histogram in the expansion order if a variable-length formulation of the SSE is used. Whenever the update method suffers from slowdown, such as at a first-order phase transition, the feedback algorithm shifts weight towards the expansion orders in the transition region, resulting in a non-uniform histogram.
It has been shown that the Metropolis algorithm can be implemented on quantum computers in a way that avoids the sign problem. However, flat histogram techniques are often preferred as they don't suffer from the same limitations that afflict Metropolis for problems of real-world interest and provide a host of other benefits. In particular, the Wang-Landau method is known for its efficiency and accuracy. In this work we design, implement, and validate a quantum algorithm for Wang-Landau sampling, greatly expanding the range of quantum many body problems solvable by Monte Carlo simulation.
We present a generalization of the classical Wang-Landau algorithm [Phys. Rev. Lett. 86, 2050 (2001)]] to quantum systems. The algorithm proceeds by stochastically evaluating the coefficients of a high temperature series expansion or a finite temperature perturbation expansion to arbitrary order. Similar to their classical counterpart, the algorithms are efficient at thermal and quantum phase transitions, greatly reducing the tunneling problem at first order phase transitions, and allow the direct calculation of the free energy and entropy.
In this communication, the histogram-reweighting technique is extended to quantum Monte Carlo studies of spin systems. The proposed method is applied to obtain equilibrium critical parameters of the quantum S=1/2S=1/2S=1/2 Heisenberg ferromagnet on simple cubic lattices. Our estimates are kBTc/J=1.67764(2),{k}_{B}{T}_{c}/J=1.67764(2),kB​Tc​/J=1.67764(2), U∗=0.4504(1),{U}^{*}=0.4504(1),U∗=0.4504(1), $\ensuremath{\gamma}=1.397(20),$ $\ensuremath{\beta}=0.360(6),$ and $\ensuremath{\nu}=0.705(10)$, which are in perfect agreement with the best field-theory and high-temperature-series results. The method can readily be applied to general quantum-spin models, standing as an important tool for the study of quantum phase transitions.
We present a general method to compute canonical averages for physical models sampled via quantum or classical quadratic unconstrained binary optimization (QUBO). First, we introduce a histogram reweighting scheme applicable to QUBO-based sampling constrained to specific intervals of an order parameter, e.g., physical energy. Next, we demonstrate that the scheme can accurately recover the density of states, which in turn allows for calculating expectation values in the conjugate ensemble, e.g., at a fixed temperature. The method can thus be used to advance the state-of-the-art characterization of physical systems that admit a QUBO-based representation and that are otherwise intractable with real-space sampling methods. A case in point are space-filling melts of lattice ring polymers, recently mapped in QUBO form, for which our method reveals that the ring catenation probability is nonmonotonic with the bending rigidity.