In this review we discuss, from a unified point of view, a variety of Monte Carlo methods used to solve eigenvalue problems in statistical mechanics and quantum mechanics. Although the applications of these methods differ widely, the underlying mathematics is quite similar in that they are stochastic implementations of the power method. In all cases, optimized trial states can be used to reduce the errors of Monte Carlo estimates.
In this review we discuss, from a unified point of view, a variety of Monte Carlo methods used to solve eigenvalue problems in statistical mechanics and quantum mechanics. Although the applications of these methods differ widely, the underlying mathematics is quite similar in that they are stochastic implementations of the power method. In all cases, optimized trial states can be used to reduce the errors of Monte Carlo estimates.
A quantum Monte Carlo method is simply a Monte Carlo method applied to a quantum problem. What distinguishes a quantum Monte Carlo method from a classical one is the initial effort necessary to represent the quantum problem in a form that is suitable for Monte Carlo simulation. It is in making this transformation that the quantum nature of the problem asserts itself not only through such obvious issues as the noncommutivity of the physical variables and the need to symmetrize or antisymmetrize the wave function, but also through less obvious issues such as the sign problem. Almost always, the transformation replaces the quantum degrees of freedom by classical ones, and it is to these classical degrees of freedom that the Monte Carlo method is actually applied. Succeeding chapters present and explain many of the quantum Monte Carlo methods being successfully used on a variety of quantum problems. In Chapters 1 and 2 we focus on discussing what the Monte Carlo method is and why it is useful. The Monte Carlo method The Monte Carlo method is not a specific technique but a general strategy for solving problems too complex to solve analytically or too intensive numerically to solve deterministically. Often a specific strategy incorporates several different Monte Carlo techniques. In what is likely the first journal article to use the phrase “Monte Carlo,” Metropolis and Ulam (1949) discuss this strategy. To paraphrase them, The Monte Carlo method is an iterative stochastic procedure, consistent with a defining relation for some function, which allows an estimate of the function without completely determining it. This is quite different from the colloquialism, “a method that uses random numbers.” Let us examine the definition piece by piece. A key point will emerge. Ulam and Metropolis were presenting the motivation and a general description of a statistical approach to the study of differential and integro-differential equations. These equations were their “defining relation for some function.” The “function” was the solution of these equations. This function is of course unknown a priori. Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller (1953) a few years later would propose a statistical approach to the study of equilibrium statistical mechanics. The defining relation there was a thermodynamic average of a physical quantity over the Boltzmann distribution. The function was the physical quantity, and the unknown its average.
Quantum Monte Carlo versions of the power method for finding ground states come with different names, including the projector method, diffusion Monte Carlo, and Green's function Monte Carlo. These quantum Monte Carlo methods are primarily adoptions of methods developed for classical problems. We summarize the basics of deterministic power methods, detail key features of Monte Carlo power methods, and put these concepts into the context of quantum ground state calculations. We conclude the chapter by outlining power methods that allow the computation of a few excited states. In the computation of excited states, we encounter sign problems, which are discussed inmore detail in Chapter 11. Themethods and techniques of the present chapter are very general and most usefully applied to systems of Bosons and to systems of Fermions and quantum spins that do not suffer from a sign problem. Deterministic direct and inverse power methods The power method is over a century old. As a deterministic method, it originally was combined with the deflation technique (Meyer, 2000; Stewart, 2001b) as a method to compute all the eigenvalues and eigenvectors of small matrices. Today, its deterministic use is limited primarily to special applications involving very large, sparse matrices. Many of these applications are in the study of quantum lattice models as the corresponding Hamiltonian matrices are typically very sparse. In these applications, we can perhaps initially store in computer memory the relatively small number of nonzero matrix elements and all the vectors. Soon, because of what is undoubtedly the now familiar exponentially increasing number of basis states that determines the order of the Hamiltonian matrix and hence the size of our vectors, we reach the point where computer memory restrictions allow us to store only a few vectors and we need to compute the matrix elements on the fly. When we have problems for which we cannot even store all the components of one vector, the power method's expression as a Monte Carlo procedure is our only option. With it, we can compute a few of the extremal eigenvalues, that is, a few of the largest or smallest eigenvalues. The mathematical basis for the method is simple.
Projection methods such as Green's function and diffusion Monte Carlo are commonly used to calculate the leading eigenvalue and eigenvector of operators or large matrices. They thereby give access to ground state properties of quantum systems, and finite temperature properties of classical statistical mechanical systems having a transfer matrix. The basis of these approaches is a stochastic application of the power method in which a "projection" operator is applied iteratively. For the systematic errors to be small, the number of iterations must be large; however, in that limit, the statistical errors grow tremendously. We present an analytical study of the main variance reduction methods used for dealing with this problem. In particular, we discuss the consequences of guiding, replication, and population control on statistical and systematic errors.
Solving the Schrodinger equation and finding excited states for quantum mechanical many-body systems is a fundamental problem. If this problem is formulated in terms of integrals, the dimensionality of the configuration space is often too high to perform the integrations required directly. This is because the volume of configuration space increases exponentially with dimension quickly, making its complete exploration impossible. Monte Carlo methods provide a way to estimate these integrals by statistically sampling a subset of configuration space, and these methods provide 1/√ N convergence regardless of dimension. For many problems involving the quantum mechanics of molecules there exists a large time scale separation between the high-frequency internal vibrations, and low-frequency intermolecular motions. This separation motivates the rigid-body approximation which freezes internal degrees of freedom in order to study intermolecular effects. The reduction in the dimensionality of configuration space further increases the range of accessible problems. This thesis is devoted to the construction and implementation of algorithms which incorporate the rigid-body approximation into existing Monte Carlo methods for solving the quantum mechanical many-body problem. Monte Carlo estimators are constructed as averages over samples drawn from some probability distribution. If this distribution is known in closed form the samples can be generated by application of the Metropolis algorithm. In most cases the distribution is not known in closed form or even representable with a finite number of variables. If this distribution is the dominant eigenstate of some known operator then a stochastic implementation of the power method can be used to generate the required samples. For the quantum mechanical problem this operator can be taken to be the imaginary-time evolution operator and its application can be represented in terms of random walks. For rigid bodies, this method involves the implementation of rotational Brownian motion. The use of quaternion algebra to implement this rotational motion enhances simplicity, performance, and numerical stability. The method described above can be generalized to investigate excited state properties using correlation function Monte Carlo which is a Monte Carlo implementation of the Rayleigh-Ritz variational method. This generalization requires the construction of a trial subspace which is then subjected to the variational method resulting in a generalized eigenvalue problem. It is important to represent optimized trial states accurately and efficiently. For this purpose, we write trial states as functions of invariant polynomials of the interparticle distances. By reducing the number of free interparticle distances, the rigid-body approximation greatly simplifies
This Perspective focuses on the several overlaps between quantum algorithms and Monte Carlo methods in the domains of physics and chemistry. We will analyze the challenges and possibilities of integrating established quantum Monte Carlo solutions into quantum algorithms. These include refined energy estimators, parameter optimization, real and imaginary-time dynamics, and variational circuits. Conversely, we will review new ideas for utilizing quantum hardware to accelerate the sampling in statistical classical models, with applications in physics, chemistry, optimization, and machine learning. This review aims to be accessible to both communities and intends to foster further algorithmic developments at the intersection of quantum computing and Monte Carlo methods. Most of the works discussed in this Perspective have emerged within the last two years, indicating a rapidly growing interest in this promising area of research.
This thesis work deals with Monte Carlo (MC) simulations with the focus on quantum mechanical systems. I start with an introduction about Monte-Carlo methods in general and their application to the classical Ising model. The second, most important part deals with the problem in quantum mechanics, and presents a possible simulation method, the Stochastic Series Expansion (SSE). I give an introduction about the idea behind that method and give some results obtained for simulations for finite spin chains. All programming was performed in FORTRAN.