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 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.
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 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 .
Introduction In Chapters 1 to 4 we studied methods for solving the Schrödinger equation for many-electron systems. Many of the techniques described there carry over to other quantum many-particle systems, such as liquid helium, and the protons and neutrons in a nucleus. The techniques which we discussed there were, however, all of a mean-field type and therefore correlation effects could not be taken into account without introducing approximations. In this chapter, we consider more accurate techniques, which are similar to those studied in Chapter 10 and are based on using (pseudo-)random numbers – hence the name ‘Monte Carlo’ for these methods. In Chapter 10 we applied Monte Carlo techniques to classical many-particle systems; here we use these techniques for studying quantum problems involving many particles. In the next section we shall see how we can apply Monte Carlo techniques to the problem of calculating the quantum mechanical expectation value of the ground state energy. This is used in order to optimise this expectation value by adjusting a trial wave function in a variational type of approach, hence the name variational Monte Carlo (VMC). In the following section we use the similarity between the Schrödinger equation and the diffusion equation in order to calculate the properties of a collection of interacting quantum mechanical particles by simulating a classical particle diffusion process. The resulting method is called diffusion Monte Carlo (DMC).
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.
… as suggested by Fermi, the time-independent Schrödinger equation … can be interpreted as describing the behavior of a system of particles each of which performs a random walk, i.e., diffuses isotropically and at the same time is subject to multiplication, which is determined by the value of the point function V. N. Metropolis and S. Ulam, 1949 Summary In the projector quantum Monte Carlo method, one uses a function of the hamiltonian to sample a distribution proportional to the exact ground-state wavefunction, and thereby computes exact matrix elements of it. An importance sampling transformation makes the algorithm much more efficient. In this chapter we introduce and develop the diffusion Monte Carlo method, which involves drifting, branching random walks. For any excited state, including any system with more than two electrons, one encounters the sign problem, limiting the direct application of these algorithms for most fermion systems. Instead, by using approximate fixed-node or fixed-phase boundary conditions, one can achieve efficiency similar to variational Monte Carlo. We also discuss the application of the projector method in a basis of Slater determinants. In this chapter, we discuss a different quantum Monte Carlo method, projector Monte Carlo (PMC). This general method was first suggested by Fermi [1049]; see the quote at the start of this chapter by two of the inventors of the Monte Carlo method. An implementation of PMC was tried out in the early days of computing [1050]. Advances in methodology, in particular importance sampling, resulted in a significant large-scale application: the exact calculation of the ground-state properties of 256 hard-sphere bosons by Kalos, Levesque, and Verlet [1051] in 1974. Calculations for electronic systems and the fixednode approximation were introduced by Anderson [1052, 1053]. One of the most important projector MC algorithms, the diffusion Monte Carlo algorithm with importance sampling for fermions, was used to compute the correlation energy of homogeneous electron gas by Ceperley and Alder [109] in 1980; the resulting HEG correlation energy was crucial in the development of density functional calculations. Types and properties of projectors In this method, a many-body projector G ( R , R ) = Ĝ is repeatedly applied to filter out the exact many-body ground state from an initial state; the operation of the projector is carried out with a random walk, hence the name of this class of methods.
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.