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

Abstract

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 ...

Machine learning and signed particles, an alternative and efficient way to simulate quantum systems

“…In this section, we briefly introduce within the context of the signed particle formulation how the evolution of particles is carried out through the computation of the Wigner kernel. Then a previous machine learning technique is described to provide the context for the particular problem in this work. As mentioned elsewhere, the applications of machine learning to molecular and material sciences are highly dependent on the amount and quality of data used to train the model, as well as the number of hidden layers and neurons in them to achieve good generalization.…”

“…We now briefly review the previous machine learning method recently discussed in Ref. and then introduce our new technique.…”

“…Eventually, a weighted average is computed in the last layer, which represents the output of the network (see Ref. for all details).…”

“…An ANN has been applied to tackle the problem of computing the Wigner kernel rapidly and reliably in the case of the signed particle formalism. The suggested neural network circumvents a training phase via the specification of weights and biases by analytical formulas, wherewith significant reduction of both computational time and memory requirements is realized.…”

“…The advantage of reducing memory consumption is especially pronounced since the kernel is now computed on the fly by the ANN only on the cells of the discretized phase‐space which are occupied by particles (the reader is encouraged to read Ref. for a complete list of details although a short summary is provided in the next section for the sake of consistency). Although this approach represents a step forward in the simulation of time‐dependent many‐body quantum systems on affordable resources, important constraint remains: the number of hidden neurons of the network must be equal to the number of cells of the discretized configuration space.…”