Neural Network Operations and Susuki-Trotter evolution of Neural Network States

TL;DR

This paper investigates how elementary quantum operations can be directly applied to Neural Network States, enabling efficient quantum state evolution and optimization without stochastic methods, by leveraging parametrizations of neural networks.

Contribution

It introduces a parametrization of quantum operations on neural network states and proposes a step-wise projection method to approximate quantum state evolution efficiently.

Findings

Neural Network States can represent states prepared by any quantum circuit with linearly growing hidden nodes.

A universal set of quantum gates can be implemented within the neural network framework.

Proposed methods allow approximate evolution of neural network states without stochastic optimization.

Abstract

It was recently proposed to leverage the representational power of artificial neural networks, in particular Restricted Boltzmann Machines, in order to model complex quantum states of many-body systems [Science, 355(6325), 2017]. States represented in this way, called Neural Network States (NNSs), were shown to display interesting properties like the ability to efficiently capture long-range quantum correlations. However, identifying an optimal neural network representation of a given state might be challenging, and so far this problem has been addressed with stochastic optimization techniques. In this work we explore a different direction. We study how the action of elementary quantum operations modifies NNSs. We parametrize a family of many body quantum operations that can be directly applied to states represented by Unrestricted Boltzmann Machines, by just adding hidden nodes and updating the network parameters. We show that this parametrization contains a set of universal quantum gates, from which it follows that the state prepared by any quantum circuit can be expressed as a Neural Network State with a number of hidden nodes that grows linearly with the number of elementary operations in the circuit. This is a powerful representation theorem (which was recently obtained with different methods) but that is not directly useful, since there is no general and efficient way to extract information from this unrestricted description of quantum states. To circumvent this problem, we propose a step-wise procedure based on the projection of Unrestricted quantum states to Restricted quantum states. In turn, two approximate methods to perform this projection are discussed. In this way, we show that it is in principle possible to approximately optimize or evolve Neural Network States without relying on stochastic methods such as Variational Monte Carlo, which are computationally expensive.