An introduction to applied quantum mechanics in the Wigner Monte Carlo formalism

Sellier, Jean Michel; Nedjalkov, Mihail; Dimov, Ivan

doi:10.1016/j.physrep.2015.03.001

Cited by 57 publications

(63 citation statements)

References 67 publications

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“…Recently a new formulation of quantum mechanics based on the concept of particles with a sign has been suggested by the author of this work [6], which is known as the signed particle formulation (its numerical discretization is, instead, known as the Wigner Monte Carlo method [7]). Despite its recent appearance, this novel formalism has been thoroughly validated in both the single-and manybody cases, showing to be uncommonly advantageous in terms of computational resources.…”

Section: The Need For Efficient Quantum Tcadmentioning

confidence: 99%

“…While the first part has been shown to have a complexity which increases linearly with the characteristic dimensions of the problem (see [7]), the computation of the Wigner kernel represents a formidable problem in terms of computational implementation. In fact, in practice it is equivalent to a multi-dimensional integral which complexity increases exponentially when approached by means of deterministic methods.…”

Section: Postulatesmentioning

confidence: 99%

“…One starts by rewriting the kernel (2) specifically for the one-body twodimensional case, which now reads (restricted to a finite domain, see [7]):…”

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Combining neural networks and signed particles to simulate quantum systems more efficiently

Sellier¹

2018

Physica A: Statistical Mechanics and its Applications

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Recently a new formulation of quantum mechanics has been suggested which describes systems by means of ensembles of classical particles provided with a sign. This novel approach mainly consists of two steps: the computation of the Wigner kernel, a multi-dimensional function describing the effects of the potential over the system, and the field-less evolution of the particles which eventually create new signed particles in the process. Although this method has proved to be extremely advantageous in terms of computational resources -as a matter of fact it is able to simulate in a time-dependent fashion manybody systems on relatively small machines -the Wigner kernel can represent the bottleneck of simulations of certain systems. Moreover, storing the kernel can be another issue as the amount of memory needed is cursed by the dimensionality of the system. In this work, we introduce a new technique which drastically reduces the computation time and memory requirement to simulate time-dependent quantum systems which is based on the use of an appropriately tailored neural network combined with the signed particle formalism. In particular, the suggested neural network is able to compute efficiently and reliably the Wigner kernel without any training as its entire set of weights and biases is specified by analytical formulas. As a consequence, the amount of memory for quantum simulations radically drops since the kernel does not need to be stored anymore as it is now computed by the neural network itself, only on the cells of the (discretized) phase-space which are occupied by particles. As its is clearly shown in the final part of this paper, not only this novel approach drastically reduces the computational time, it also remains accurate. The author believes this work opens the way towards effective design of quantum devices, with incredible practical implications.Keywords: Quantum mechanics, Neural networks, Signed particle formulation, Wigner kernel, Quantum technology, Computer aided design Preprint submitted to ElsevierOctober 31, 2017 The need for efficient quantum TCADAbout a century ago, in order to understand a series of experiments involving small physical objects such as electrons, atoms and molecules, a peculiar theory was conceived, known today as quantum mechanics. This remarkable theoretical framework consists of a distinct set of rules which can (in a broad sense) explain and predict the observed features of what we call a quantum system. The implications of such theory are not only of philosophical importance, they also have a huge relevance in applied fields such as electronics and nanoelectronics. In fact, as today semiconductor devices have active lengths in the range of a few tens of nanometers, quantum effects are dominant and classically designed complementary metal-oxide-semiconductor (CMOS) transistors do not operate reliably any longer.Presently, we are entering in the post-CMOS era. Although this might sound like the end of a very successful period for the semiconductor industry, one ...

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Section: The Need For Efficient Quantum Tcadmentioning

confidence: 99%

Section: Postulatesmentioning

confidence: 99%

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Combining neural networks and signed particles to simulate quantum systems more efficiently

Sellier¹

2018

Physica A: Statistical Mechanics and its Applications

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“…f (x, k, t) = f i (xcos(ωt) − ksin(ωt), xsin(ωt) + kcos(ωt)) (12) This can be seen by applying the operations of (1) on (12) and taking into account that the operator with the Wigner potential of a harmonic oscillator can be equivalently formulated as a force term, which completes the left hand side of (1) to a general Liouville operator. In this case there is no difference between classical and quantum evolution, and quantum mechanics enters via the initial condition which must obey the uncertainty relations.…”

Section: Physical Aspectsmentioning

confidence: 99%

Boundary conditions and the Wigner equation solution

et al. 2015

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We consider the existence and uniqueness of the solution of the Wigner equation in the presence of boundary conditions. The equation, describing electron transport in nanostructures, is analyzed in terms of the Neumann series expansion of the corresponding integral form, obtained with the help of classical particle trajectories. It is shown that the mathematical aspects of the solution can not be separated from the physical attributes of the problem. In the presented analysis these two sides of the problem mutually interplay, which is of importance for understanding of the peculiarities of Wigner-quantum transport. The problem is first formulated as the long time limit of a general evolution process posed by initial and boundary conditions. Then the Wigner equation is reformulated as a second kind of a Fredholm integral equation which is of Volterra type with respect to the time variable. The analysis of the convergence of the corresponding Neumann series, sometimes called Liouville-Neumann series, relies on the assumption for reasonable local conditions obeyed by the kernel.

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“…Different methods have been developed to solve the WTE, including direct solutions [18,19] and particle Monte Carlo algorithms [20][21][22], in strong analogy with what has been developed in the past for solving the BTE. It has been recently extended to treat time-dependent ab initio simulations as well as many-body quantum problems and entangled systems [23,24]. The analogy between Boltzmann and Wigner formalisms is so strong that to consider the scattering effects in the Wigner approach of quantum transport it has been shown that the Boltzmann collision operator, based on the Fermi golden rule to treat instantaneous scattering events, is usually a very good approximation [25,26].…”

Section: Introductionmentioning

confidence: 99%

Introduction

Bescond

Dollfus

2016

J Comput Electron

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The continuous downscaling of electronic devices has led to remarkable broadening of applications during the last twenty years, from nanoelectronics through the photovoltaics and more recently quantum computing.Although these devices have reached nanodimensions, with carrier transport approaching the ballistic limit in the active region, it is widely recognized that inelastic scattering have still a significant influence. For instance, it has been shown theoretically that electron-phonon scattering drastically influences the current characteristics in ultimate nanowire transistors [1]. The miniaturization tends to enhance also the impact of thermal effects, and accurate description of phonon scattering at the nanoscale is becoming crucial to understand and manage thermal dissipation in devices and circuits [2]. In modern optoelectronic devices like new generation solar cells, it is also essential to consider appropriate models of light-matter interaction at the microscopic level [3,4]. Time-dependent quantum transport models are becoming more and more relevant to correctly assess and predict the properties of electronic quantum computing devices. In such devices, the energy of an electron can change during its travel across the active region and timedependent modeling techniques are essential to predict their physical behavior [5]. B Philippe Dollfus

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An introduction to applied quantum mechanics in the Wigner Monte Carlo formalism

Cited by 57 publications

References 67 publications

Combining neural networks and signed particles to simulate quantum systems more efficiently

Combining neural networks and signed particles to simulate quantum systems more efficiently

Boundary conditions and the Wigner equation solution

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