In computational physics, **variational Monte Carlo (VMC)** is a quantum Monte Carlo method that applies the variational method to approximate the ground state of a quantum system.^{ [1] }

- Wave function optimization in VMC
- VMC and deep learning
- See also
- Further reading
- General
- Wave-function optimization in VMC
- References

The basic building block is a generic wave function depending on some parameters . The optimal values of the parameters is then found upon minimizing the total energy of the system.

In particular, given the Hamiltonian , and denoting with a many-body configuration, the expectation value of the energy can be written as:^{ [2] }

Following the Monte Carlo method for evaluating integrals, we can interpret as a probability distribution function, sample it, and evaluate the energy expectation value as the average of the so-called local energy . Once is known for a given set of variational parameters , then optimization is performed in order to minimize the energy and obtain the best possible representation of the ground-state wave-function.

VMC is no different from any other variational method, except that the many-dimensional integrals are evaluated numerically. Monte Carlo integration is particularly crucial in this problem since the dimension of the many-body Hilbert space, comprising all the possible values of the configurations , typically grows exponentially with the size of the physical system. Other approaches to the numerical evaluation of the energy expectation values would therefore, in general, limit applications to much smaller systems than those analyzable thanks to the Monte Carlo approach.

The accuracy of the method then largely depends on the choice of the variational state. The simplest choice typically corresponds to a mean-field form, where the state is written as a factorization over the Hilbert space. This particularly simple form is typically not very accurate since it neglects many-body effects. One of the largest gains in accuracy over writing the wave function separably comes from the introduction of the so-called Jastrow factor. In this case the wave function is written as , where is the distance between a pair of quantum particles and is a variational function to be determined. With this factor, we can explicitly account for particle-particle correlation, but the many-body integral becomes unseparable, so Monte Carlo is the only way to evaluate it efficiently. In chemical systems, slightly more sophisticated versions of this factor can obtain 80–90% of the correlation energy (see electronic correlation) with less than 30 parameters. In comparison, a configuration interaction calculation may require around 50,000 parameters to reach that accuracy, although it depends greatly on the particular case being considered. In addition, VMC usually scales as a small power of the number of particles in the simulation, usually something like *N*^{2−4} for calculation of the energy expectation value, depending on the form of the wave function.

QMC calculations crucially depend on the quality of the trial-function, and so it is essential to have an optimized wave-function as close as possible to the ground state. The problem of function optimization is a very important research topic in numerical simulation. In QMC, in addition to the usual difficulties to find the minimum of multidimensional parametric function, the statistical noise is present in the estimate of the cost function (usually the energy), and its derivatives, required for an efficient optimization.

Different cost functions and different strategies were used to optimize a many-body trial-function. Usually three cost functions were used in QMC optimization energy, variance or a linear combination of them. The variance optimization method has the advantage that the exact wavefunction's variance is known. (Because the exact wavefunction is an eigenfunction of the Hamiltonian, the variance of the local energy is zero). This means that variance optimization is ideal in that it is bounded by below, it is positive defined and its minimum is known. Energy minimization may ultimately prove more effective, however, as different authors recently showed that the energy optimization is more effective than the variance one.

There are different motivations for this: first, usually one is interested in the lowest energy rather than in the lowest variance in both variational and diffusion Monte Carlo; second, variance optimization takes many iterations to optimize determinant parameters and often the optimization can get stuck in multiple local minimum and it suffers of the "false convergence" problem; third energy-minimized wave functions on average yield more accurate values of other expectation values than variance minimized wave functions do.

The optimization strategies can be divided into three categories. The first strategy is based on correlated sampling together with deterministic optimization methods. Even if this idea yielded very accurate results for the first-row atoms, this procedure can have problems if parameters affect the nodes, and moreover density ratio of the current and initial trial-function increases exponentially with the size of the system. In the second strategy one use a large bin to evaluate the cost function and its derivatives in such way that the noise can be neglected and deterministic methods can be used.

The third approach, is based on an iterative technique to handle directly with noise functions. The first example of these methods is the so-called Stochastic Gradient Approximation (SGA), that was used also for structure optimization. Recently an improved and faster approach of this kind was proposed the so-called Stochastic Reconfiguration (SR) method.

In 2017, Giuseppe Carleo and Matthias Troyer ^{ [3] } used a VMC objective function to train an artificial neural network to find the ground state of a quantum mechanical system. More generally, artificial neural networks are being used as a wave function ansatz (known as neural network quantum states) in VMC frameworks for finding ground states of quantum mechanical systems. The use of neural network ansatzes for VMC has been extended to fermions, enabling electronic structure calculations that are significantly more accurate than VMC calculations which do not use neural networks.^{ [4] }^{ [5] }^{ [6] }

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**Quantum mechanics** is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science.

In quantum mechanics, the **uncertainty principle** is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, *x*, and momentum, *p*, can be predicted from initial conditions.

The **de Broglie–Bohm theory**, also known as the *pilot wave theory*, **Bohmian mechanics**, **Bohm's interpretation**, and the **causal interpretation**, is an interpretation of quantum mechanics. In addition to the wavefunction, it also postulates an actual configuration of particles exists even when unobserved. The evolution over time of the configuration of all particles is defined by a guiding equation. The evolution of the wave function over time is given by the Schrödinger equation. The theory is named after Louis de Broglie (1892–1987) and David Bohm (1917–1992).

**Loop quantum gravity** (**LQG**) is a theory of quantum gravity, which aims to merge quantum mechanics and general relativity, incorporating matter of the Standard Model into the framework established for the pure quantum gravity case. It is an attempt to develop a quantum theory of gravity based directly on Einstein's geometric formulation rather than the treatment of gravity as a force. As a theory LQG postulates that the structure of space and time is composed of finite loops woven into an extremely fine fabric or network. These networks of loops are called spin networks. The evolution of a spin network, or spin foam, has a scale above the order of a Planck length, approximately 10^{−35} meters, and smaller scales are meaningless. Consequently, not just matter, but space itself, prefers an atomic structure.

In quantum mechanics, **einselections**, short for "environment-induced superselection", is a name coined by Wojciech H. Zurek for a process which is claimed to explain the appearance of wavefunction collapse and the emergence of classical descriptions of reality from quantum descriptions. In this approach, classicality is described as an emergent property induced in open quantum systems by their environments. Due to the interaction with the environment, the vast majority of states in the Hilbert space of a quantum open system become highly unstable due to entangling interaction with the environment, which in effect monitors selected observables of the system. After a decoherence time, which for macroscopic objects is typically many orders of magnitude shorter than any other dynamical timescale, a generic quantum state decays into an uncertain state which can be expressed as a mixture of simple pointer states. In this way the environment induces effective superselection rules. Thus, einselection precludes stable existence of pure superpositions of pointer states. These 'pointer states' are stable despite environmental interaction. The einselected states lack coherence, and therefore do not exhibit the quantum behaviours of entanglement and superposition.

In theoretical physics, the **pilot wave theory**, also known as **Bohmian mechanics**, was the first known example of a hidden-variable theory, presented by Louis de Broglie in 1927. Its more modern version, the de Broglie–Bohm theory, interprets quantum mechanics as a deterministic theory, avoiding troublesome notions such as wave–particle duality, instantaneous wave function collapse, and the paradox of Schrödinger's cat. To solve these problems, the theory is inherently nonlocal.

The **Wheeler–DeWitt equation** for theoretical physics and applied mathematics, is a field equation attributed to John Archibald Wheeler and Bryce DeWitt. The equation attempts to mathematically combine the ideas of quantum mechanics and general relativity, a step towards a theory of quantum gravity.

**Quantum Monte Carlo** encompasses a large family of computational methods whose common aim is the study of complex quantum systems. One of the major goals of these approaches is to provide a reliable solution of the quantum many-body problem. The diverse flavors of quantum Monte Carlo approaches all share the common use of the Monte Carlo method to handle the multi-dimensional integrals that arise in the different formulations of the many-body problem.

**Car–Parrinello molecular dynamics** or **CPMD** refers to either a method used in molecular dynamics or the computational chemistry software package used to implement this method.

**Diffusion Monte Carlo** (DMC) or **diffusion quantum Monte Carlo** is a quantum Monte Carlo method that uses a Green's function to solve the Schrödinger equation. DMC is potentially numerically exact, meaning that it can find the exact ground state energy within a given error for any quantum system. When actually attempting the calculation, one finds that for bosons, the algorithm scales as a polynomial with the system size, but for fermions, DMC scales exponentially with the system size. This makes exact large-scale DMC simulations for fermions impossible; however, DMC employing a clever approximation known as the fixed-node approximation can still yield very accurate results.

The **Ghirardi–Rimini–Weber theory** (**GRW**) is a spontaneous collapse theory in quantum mechanics, proposed in 1986 by Giancarlo Ghirardi, Alberto Rimini, and Tullio Weber.

In applied mathematics, the **numerical sign problem** is the problem of numerically evaluating the integral of a highly oscillatory function of a large number of variables. Numerical methods fail because of the near-cancellation of the positive and negative contributions to the integral. Each has to be integrated to very high precision in order for their difference to be obtained with useful accuracy.

**David Matthew Ceperley** is a theoretical physicist in the physics department at the University of Illinois Urbana-Champaign or UIUC. He is a world expert in the area of Quantum Monte Carlo computations, a method of calculation that is generally recognised to provide accurate quantitative results for many-body problems described by quantum mechanics.

In condensed matter physics, **biexcitons** are created from two free excitons.

In theoretical physics, the **logarithmic Schrödinger equation** is one of the nonlinear modifications of Schrödinger's equation. It is a classical wave equation with applications to extensions of quantum mechanics, quantum optics, nuclear physics, transport and diffusion phenomena, open quantum systems and information theory, effective quantum gravity and physical vacuum models and theory of superfluidity and Bose–Einstein condensation. Its relativistic version was first proposed by Gerald Rosen. It is an example of an integrable model.

The **Koopman–von Neumann mechanics** is a description of classical mechanics in terms of Hilbert space, introduced by Bernard Koopman and John von Neumann in 1931 and 1932, respectively.

The light front quantization of quantum field theories provides a useful alternative to ordinary equal-time quantization. In particular, it can lead to a relativistic description of bound systems in terms of quantum-mechanical wave functions. The quantization is based on the choice of light-front coordinates, where plays the role of time and the corresponding spatial coordinate is . Here, is the ordinary time, is one Cartesian coordinate, and is the speed of light. The other two Cartesian coordinates, and , are untouched and often called transverse or perpendicular, denoted by symbols of the type . The choice of the frame of reference where the time and -axis are defined can be left unspecified in an exactly soluble relativistic theory, but in practical calculations some choices may be more suitable than others.

The **time-dependent variational Monte Carlo (t-VMC)** method is a quantum Monte Carlo approach to study the dynamics of closed, non-relativistic quantum systems in the context of the quantum many-body problem. It is an extension of the variational Monte Carlo method, in which a time-dependent pure quantum state is encoded by some variational wave function, generally parametrized as

In nuclear physics, **ab initio methods** seek to describe the atomic nucleus from the bottom up by solving the non-relativistic Schrödinger equation for all constituent nucleons and the forces between them. This is done either exactly for very light nuclei or by employing certain well-controlled approximations for heavier nuclei. Ab initio methods constitute a more fundamental approach compared to e.g. the nuclear shell model. Recent progress has enabled ab initio treatment of heavier nuclei such as nickel.

**Neural Network Quantum States** is a general class of variational quantum states parameterized in terms of an artificial neural network. It was first introduced in 2017 by the physicists Giuseppe Carleo and Matthias Troyer to approximate wave functions of many-body quantum systems.

- ↑ Scherer, Philipp O.J. (2017).
*Computational Physics*. Graduate Texts in Physics. Cham: Springer International Publishing. doi:10.1007/978-3-319-61088-7. ISBN 978-3-319-61087-0. - ↑ Kalos, Malvin H., ed. (1984).
*Monte Carlo Methods in Quantum Problems*. Dordrecht: Springer Netherlands. doi:10.1007/978-94-009-6384-9. ISBN 978-94-009-6386-3. - ↑ Carleo, Giuseppe; Troyer, Matthias (2017). "Solving the Quantum Many-Body Problem with Artificial Neural Networks".
*Science*.**355**(6325). arXiv: 1606.02318 . doi:10.1126/science.aag2302. - ↑ Pfau, David; Spencer, James; Matthews, Alexander G. de G.; Foulkes, W. M. C. (2020). "Ab-initio Solution of the Many-Electron Schrödinger Equation with Deep Neural Networks".
*Physical Review Research*.**2**(3). arXiv: 1909.02487 . doi:10.1103/PhysRevResearch.2.033429. - ↑ Hermann, Jan; Schätzle, Zeno; Noé, Frank (2020). "Deep Neural Network Solution of the Electronic Schrödinger Equation".
*Nature Chemistry*.**12**. arXiv: 1909.08423 . doi:10.1038/s41557-020-0544-y. - ↑ Choo, Kenny; Mezzacapo, Antonio; Carleo, Giuseppe (2020). "Fermionic Neural-Network States for Ab-initio Electronic Structure".
*Nature Communications*.**11**. arXiv: 1909.12852 . doi:10.1038/s41467-020-15724-9. PMC 7217823 .

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