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arXiv:1806.00082

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

Jean Michel Sellier, Jacob Leygonie, Gaétan Marceau

Published May 29, 2018• 0 citations
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Abstract

Recently the use of neural networks has been introduced in the context of the signed particle formulation of quantum mechanics to rapidly and reliably compute the Wigner kernel of any provided potential. This new technique has introduced two important advantages over the more standard finite difference/element methods: 1) it reduces the amount of memory required for the simulation of a quantum system. As a matter of fact, it does not require storing the kernel in a (expensive) multi-dimensional array, and 2) a consistent speedup is obtained since now one can compute the kernel on the cells of interest only, i.e. the cells occupied by signed particles. Although this certainly represents a step forward into the direction of rapid simulations of quantum systems, it comes at a price: the number of hidden neurons is constrained by design to be equal to the number of cells of the discretized real space. It is easy to see how this limitation can quickly become an issue when very fine meshes are necessary. In this work, we continue to ameliorate this previous approach by reducing the complexity of the neural network and, consequently, by introducing an additional speedup. More specifically, we propose a new network architecture which requires less neurons than the previous approach in its hidden layer. For validation purposes, we apply this novel technique to a well known simple, but very indicative, one-dimensional quantum system consisting of a Gaussian wave packet interacting with a potential barrier. In order to clearly show the validity of our suggested approach, time-dependent comparisons with the previous technique are presented. In spite of its simpler architecture, a good agreement is observed, thus representing one step further towards fast and reliable simulations of time-dependent quantum systems.

DOI
10.48550/arxiv.1806.00082

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Recently the use of neural networks has been introduced in the context of the signed particle formulation of quantum mechanics to rapidly and reliably compute the Wigner kernel of any provided potential. This new technique has introduced two important advantages over the more standard finite difference/element methods: 1) it reduces the amount of memory required for the simulation of a quantum system. As a matter of fact, it does not require storing the kernel in a (expensive) multi-dimensional array, and 2) a consistent speedup is obtained since now one can compute the kernel on the cells of interest only, i.e. the cells occupied by signed particles. Although this certainly represents a step forward into the direction of rapid simulations of quantum systems, it comes at a price: the number of hidden neurons is constrained by design to be equal to the number of cells of the discretized real space. It is easy to see how this limitation can quickly become an issue when very fine meshes are necessary. In this work, we continue to ameliorate this previous approach by reducing the complexity of the neural network and, consequently, by introducing an additional speedup. More specifically, we propose a new network architecture which requires less neurons than the previous approach in its hidden layer. For validation purposes, we apply this novel technique to a well known simple, but very indicative, one-dimensional quantum system consisting of a Gaussian wave packet interacting with a potential barrier. In order to clearly show the validity of our suggested approach, time-dependent comparisons with the previous technique are presented. In spite of its simpler architecture, a good agreement is observed, thus representing one step further towards fast and reliable simulations of time-dependent quantum systems.
Abstract Recently neural networks have been applied in the context of the signed particle formulation of quantum mechanics to rapidly and reliably compute the Wigner kernel of any provided potential. Important advantages were introduced, such as the reduction of the amount of memory required for the simulation of a quantum system by avoiding the storage of the kernel in a multi‐dimensional array, as well as attainment of consistent speedup by the ability to realize the computation only on the cells occupied by signed particles. An inherent limitation was the number of hidden neurons to be equal to the number of cells of the discretized real space. In this work, anew network architecture is presented, decreasing the number of neurons in its hidden layer, thereby reducing the complexity of the network and achieving an additional speedup. The approach is validated on a onedimensional quantum system consisting of a Gaussian wave packet interacting with a potential barrier.
This work belongs to a series of articles which have been dedicated to the combination of signed particles and neural networks to speed up the time-dependent simulation of quantum systems. More specifically, the suggested networks are utilized to compute the function known as the Wigner kernel. In the first paper, we suggested a network which is completely defined analytically and which does not necessitate any training process. Although very useful, this approach keeps the same complexity as the more standard finite difference methods. In the second work, we presented a different architecture which has generalization capabilities. Although more convenient in terms of computational time, it still uses a similar structure compared to the previous approach, with less neurons in the hidden layer but with the same expensive activation functions (sine functions). In this work, we focus on neural networks without any prior physics based knowledge. In more details, the network now consists of different hidden layers which are based on more common, and less computationally expensive, activation functions such as rectified linear units, and which focus on predicting only one column of the discretized Wigner kernel at a time. This new suggested architecture proves to accurately learn the transform from the space of potential functions to the space of Wigner kernels and performs very well during the simulation of quantum systems (20 times faster). Finally, we simulate a one-dimensional quantum system consisting of a wave packet impinging on a potential barrier for validation purposes. This represents one further step ahead to achieve fast and reliable simulations of time-dependent quantum systems.
Obtaining accurate properties of many-body interacting quantum matter is a long-standing challenge in theoretical physics and chemistry, rooting into the complexity of the many-body wave-function. Classical representations of many-body states constitute a key tool for both analytical and numerical approaches to interacting quantum problems. Here, we introduce a technique to construct classical representations of many-body quantum systems based on artificial neural networks. Our constructions are based on the deep Boltzmann machine architecture, in which two layers of hidden neurons mediate quantum correlations. The approach reproduces the exact imaginary-time evolution for many-body lattice Hamiltonians, is completely deterministic, and yields networks with a polynomially-scaling number of neurons. We provide examples where physical properties of spin Hamiltonians can be efficiently obtained. Also, we show how systematic improvements upon existing restricted Boltzmann machines ansatze can be obtained. Our method is an alternative to the standard path integral and opens new routes in representing quantum many-body states.
Abstract Obtaining accurate properties of many-body interacting quantum matter is a long-standing challenge in theoretical physics and chemistry, rooting into the complexity of the many-body wave-function. Classical representations of many-body states constitute a key tool for both analytical and numerical approaches to interacting quantum problems. Here, we introduce a technique to construct classical representations of many-body quantum systems based on artificial neural networks. Our constructions are based on the deep Boltzmann machine architecture, in which two layers of hidden neurons mediate quantum correlations. The approach reproduces the exact imaginary-time evolution for many-body lattice Hamiltonians, is completely deterministic, and yields networks with a polynomially-scaling number of neurons. We provide examples where physical properties of spin Hamiltonians can be efficiently obtained. Also, we show how systematic improvements upon existing restricted Boltzmann machines ansatze can be obtained. Our method is an alternative to the standard path integral and opens new routes in representing quantum many-body states.
Computing dynamical distributions in quantum many-body systems represents one of the paradigmatic open problems in theoretical condensed matter physics. Despite the existence of different techniques both in real-time and frequency space, computational limitations often dramatically constrain the physical regimes in which quantum many-body dynamics can be efficiently solved. Here we show that the combination of machine-learning methods and complementary many-body tensor network techniques substantially decreases the computational cost of quantum many-body dynamics. We demonstrate that combining kernel polynomial techniques and real-time evolution, together with deep neural networks, allows to compute dynamical quantities faithfully. Focusing on many-body dynamical distributions, we show that this hybrid neural-network many-body algorithm, trained with single-particle data only, can efficiently extrapolate dynamics for many-body systems without prior knowledge. Importantly, this algorithm is shown to be substantially resilient to numerical noise, a feature of major importance when using this algorithm together with noisy many-body methods. Ultimately, our results provide a starting point towards neural-network powered algorithms to support a variety of quantum many-body dynamical methods, that could potentially solve computationally expensive many-body systems in a more efficient manner.
We compute the ground-state properties of fully polarized, trapped, one-dimensional fermionic systems interacting through a Gaussian potential. We use an antisymmetric artificial neural network, or neural quantum state, as an Ansatz for the wave function and use machine learning techniques to variationally minimize the energy of systems from two to six particles. We provide extensive benchmarks for this toy model with other many-body methods, including exact diagonalization and the Hartree-Fock approximation. The neural quantum state provides the best energies across a wide range of interaction strengths. We find very different ground states depending on the sign of the interaction. In the nonperturbative repulsive regime, the system asymptotically reaches crystalline order. In contrast, the strongly attractive regime shows signs of bosonization. The neural quantum state continuously learns these different phases with an almost constant number of parameters and a very modest increase in computational time with the number of particles.