In this brief article, I introduce the various types of quantum Monte Carlo (QMC) methods, in particular, those that are applicable to systems in extreme regimes of temperature and pressure.We give references to longer articles where detailed discussion of applications and algorithms appear. MOTIVATIONOne does QMC for the same reason as one does classical simulations; there is no other method able to treat exactly the quantum many-body problem aside from the direct simulation method where electrons and ions are directly represented as particles, instead of as a "fluid" as is done in mean-field based methods.However, quantum systems are more difficult than classical systems because one does not know the distribution to be sampled, it must be solved for.In fact, it is not known today which quantum problems can be "solved" with simulation on a classical computer in a reasonable amount of computer time.One knows that certain systems, such as most quantum many-body systems in 1D and most bosonic systems are amenable to solution with Monte Carlo methods, however, the "sign problem" prevents making the same statement for systems with electrons in 3D.Some limitations or approximations are needed in practice.On the other hand, in contrast to simulation of classical systems, one does know the Hamiltonian exactly: namely charged particles interacting with a Coulomb potential.Given sufficient computer resources, the results can be of quite high quality and for systems where there is little reliable experimental data.For this reason, QMC methods, though more expensive, are useful to benchmark and validate results from other methods.The two main applications that we will discuss in this review are the "electronic structure problem"; computing the energy of the an interacting systems of electrons and fixed ions, and the problem of quantum effects of the nuclei. RANDOM WALK AND MARKOV CHAINSMost QMC algorithms are based on random walks, known mathematically as Markov chains.This is a general class of algorithms, introduced in the famous work by Metropolis, the Rosenbluths and the Tellers (1953) for sampling any probability distribution.Let us denote the desired distribution as Π(S), where the variable "S" represents the state of the walk, for example the coordinates of all of the particles.In a random walk, one invents some moving rules (transition probabilities) for changing the state of the system so that the probability of being in state S n+1 in the (n+1) step, given that the random walk was S n is given by T(S n ; S n+1 ).By
In this brief article, I introduce the various types of quantum Monte Carlo (QMC) methods, in particular, those that are applicable to systems in extreme regimes of temperature and pressure.We give references to longer articles where detailed discussion of applications and algorithms appear. MOTIVATIONOne does QMC for the same reason as one does classical simulations; there is no other method able to treat exactly the quantum many-body problem aside from the direct simulation method where electrons and ions are directly represented as particles, instead of as a "fluid" as is done in mean-field based methods.However, quantum systems are more difficult than classical systems because one does not know the distribution to be sampled, it must be solved for.In fact, it is not known today which quantum problems can be "solved" with simulation on a classical computer in a reasonable amount of computer time.One knows that certain systems, such as most quantum many-body systems in 1D and most bosonic systems are amenable to solution with Monte Carlo methods, however, the "sign problem" prevents making the same statement for systems with electrons in 3D.Some limitations or approximations are needed in practice.On the other hand, in contrast to simulation of classical systems, one does know the Hamiltonian exactly: namely charged particles interacting with a Coulomb potential.Given sufficient computer resources, the results can be of quite high quality and for systems where there is little reliable experimental data.For this reason, QMC methods, though more expensive, are useful to benchmark and validate results from other methods.The two main applications that we will discuss in this review are the "electronic structure problem"; computing the energy of the an interacting systems of electrons and fixed ions, and the problem of quantum effects of the nuclei. RANDOM WALK AND MARKOV CHAINSMost QMC algorithms are based on random walks, known mathematically as Markov chains.This is a general class of algorithms, introduced in the famous work by Metropolis, the Rosenbluths and the Tellers (1953) for sampling any probability distribution.Let us denote the desired distribution as Π(S), where the variable "S" represents the state of the walk, for example the coordinates of all of the particles.In a random walk, one invents some moving rules (transition probabilities) for changing the state of the system so that the probability of being in state S n+1 in the (n+1) step, given that the random walk was S n is given by T(S n ; S n+1 ).By
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.
… a general method, suitable for electronic computing machines, of calculating the properties of any substance which may be considered as composed of interacting individual molecules. N. Metropolis et al. , 1953 Summary Quantum Monte Carlo methods have been very useful in providing exact results, or, at least, exact constraints on properties of electronic systems, in particular for the homogeneous electron gas. The results are, in many cases, more accurate than those from other quantum many-body methods, and provide unique capabilities and insights. In this chapter we introduce the general properties of stochastic methods and motivate their use on the quantum many-body problem. In particular, we discuss Markov chains and the computation of error estimates. The methods that we introduce in the next four chapters are quite different from those in Parts II and III: stochastic or quantum Monte Carlo methods. In stochastic methods, instead of solving deterministically for properties of the quantum many-body system, one sets up a random walk that samples for the properties. Historically the most important role of QMC for the electronic structure field has been to provide input into the other methods, most notably the QMC calculation of the HEG [109], used for the exchange– correlation functional in DFT. A second important role has been as benchmarks for other methods such asGW. There are systems for which QMC is uniquely suited, for example the Wigner transition in the low-density electron gas, see Sec. 3.1. In this chapter we introduce general properties of simulations, in particular Markov chains, and error estimates. In the following chapters we will apply this theory to three general classes of quantum Monte Carlo algorithms, namely variational (Ch. 23), projector (Ch. 24), and path-integral Monte Carlo (Ch. 25); Ch. 18 already introduced the QMC calculation of the impurity Green's function used in the dynamical mean-field method.We note that there are a variety of other QMC methods not covered in this book. Simulations First let us define what we mean by a simulation, since the word has other meanings in applied science. The dimensionality of phase space (i.e., the Hilbert space for a quantum system) is large or infinite. Even a classical system requires the positions and momenta of all particles, and, hence, the phase space for N classical particles has dimensionality 6 N .
Quantum Monte Carlo (QMC) methods represent a powerful family of computational techniques for tackling complex quantum many-body problems and performing calculations of stationary state properties. QMC is among the most accurate and powerful approaches to the study of electronic structure, but its application is often hindered by a steep learning curve; hence it is rarely addressed in undergraduate and postgraduate classes. This tutorial is a step toward filling this gap. We offer an introduction to the diffusion Monte Carlo (DMC) method, which aims to solve the imaginary time Schrödinger equation through stochastic sampling of the configuration space. Starting from the theoretical foundations, the discussion leads naturally to the formulation of a step-by-step algorithm. To illustrate how the method works in simplified scenarios, examples such as the harmonic oscillator and the hydrogen atom are provided. The discussion extends to the fixed-node approximation, a crucial approach for addressing the fermionic sign problem in multi-electron systems. In particular, we examine the influence of trial wave function nodal surfaces on the accuracy of DMC energy by evaluating results from a non-interacting two-fermion system. Extending the method to excited states is feasible in principle, but some additional considerations are needed, supported by practical insights. By addressing the fundamental concepts from a hands-on perspective, we hope this tutorial will serve as a valuable guide for researchers and students approaching DMC for the first time.
Electronic structure quantum Monte Carlo Quantum Monte Carlo (QMC) is an advanced simulation methodology for studies of manybody quantum systems. The QMC approaches combine analytical insights with stochastic computational techniques for efficient solution of several classes of important many-body problems such as the stationary Schrödinger equation. QMC methods of various flavors have been applied to a great variety of systems spanning continuous and lattice quantum models, molecular and condensed systems, BEC-BCS ultracold condensates, nuclei, etc. In this review, we focus on the electronic structure QMC, i.e., methods relevant for systems described by the electron-ion Hamiltonians. Some of the key QMC achievements include direct treatment of electron correlation, accuracy in predicting energy differences and favorable scaling in the system size. Calculations of atoms, molecules, clusters and solids have demonstrated QMC applicability to real systems with hundreds of electrons while providing 90-95% of the correlation energy and energy differences typically within a few percent of experiments. Advances in accuracy beyond these limits are hampered by the so-called fixed-node approximation which is used to circumvent the notorious fermion sign problem. Many-body nodes of fermion states and their properties have therefore become one of the important topics for further progress in predictive power and efficiency of QMC calculations. Some of our recent results on the wave function nodes and related nodal domain topologies will be briefly reviewed. This includes analysis of few-electron systems and descriptions of exact and approximate nodes using transformations and projections of the highly-dimensional nodal hypersurfaces into the 3D space. Studies of fermion nodes offer new insights into topological properties of eigenstates such as explicit demonstrations that generic fermionic ground states exhibit the minimal number of two nodal domains. Recently proposed trial wave functions based on Pfaffians with pairing orbitals are presented and their nodal properties are tested in calculations of first row atoms and molecules. Finally, backflow "dressed" coordinates are introduced as another possibility for capturing correlation effects and for decreasing the fixed-node bias.
Our discussion of Markov chains, with the exception of mentioning the Metropolis and heat-bath algorithms, has so far been very general with little contact with issues and opportunities related to specific applications. In this chapter, we recall that our target is many-body problems defined on a lattice and introduce several frameworks exploiting what is special about Markov processes for these types of problems. We consider here classical many-body problems, using the Ising model as the representative. Our discussion will be extended to various quantum many-body problems and algorithms in subsequent chapters. Many-body phase space The numerical difficulty of studying many-body problems on a lattice arises from the fact that their phase space Ω is a direct product of many phase spaces of local degrees of freedom. Generally, a local phase space is associated with each lattice site. If n is the size of this phase space and N is the number of lattice sites, then the number of states |Ω| available to the whole system is n N . In other words, the number of states in the phase space grows exponentially fast with the physical size of the system. For example, in the Ising model, the Ising spin s i on each site can take one of the two values ±1, and hence the number of states in the total phase space is 2 N . In Chapter 1, we noted that this exponential scaling thwarts deterministic solutions and is a reason why we use the Monte Carlo method. The exponentially large number of states implies that the enumeration of all states, which requires a computational effort proportional to the size of the phase space, is not an option for a problem with a large number of sites. As discussed in Section 2.7, the Monte Carlo method generally samples from a compressed phase space, avoiding the need to solve the entire problem. However, even with this advantage, can we reduce the computational effort to a manageable level? An obvious requirement is that we can equilibrate the simulation in a reasonable amount of computer time. We know that the Markov process converges to a stationary state sampling some distribution (Section 2.4), and for detailed balance algorithms, Rosenbluth's theorem (Section 2.6) guarantees monotonic convergence.
A brief review will be given about the state of the art of Quantum Monte Carlo (QMC) simulations in solid state physics. Particularly in this field, QMC has become very popular during the last couple of years due to the growing interest in reliable results for strongly correlated electronic systems. It will be demonstrated how QMC schemes cope with typical many many-body problems. Representative applications will be discussed. Present algorithms suffer, however, from various problems, restricting the practicableness of QMC. An exposition of the difficulties will be given which have to be overcome to make QMC studies applicable to a much wider range of physical problems.