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arXiv:1907.12844 · Physical review. B./Physical review. B

Sampling scheme for neuromorphic simulation of entangled quantum systems

Stefanie Czischek, Jan Martin Pawlowski, Thomas Gasenzer, Martin Gärttner

Published Nov 13, 2019• 10 citations
arXivHTMLPDF

Abstract

Neural networks can be used to represent a large class of quantum many-body states in a classical-statistical manner. This includes entangled states such as the paradigmatic Bell states or a quantum Ising chain near its critical point. The task of sampling from these states may be performed efficiently by physically implementing the network on neuromorphic hardware. A deep neural network structure is proposed for the sampling of arbitrary and, in general, noncommuting observables, where a phase-reweighting scheme is used in view of the complex coupling parameters. Entanglement is demonstrated through the violation of Bell inequalities.

DOI
10.1103/physrevb.100.195120

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Neural networks can be used to represent a large class of quantum many-body states in a classical-statistical manner. This includes entangled states such as the paradigmatic Bell states or a quantum Ising chain near its critical point. The task of sampling from these states may be performed efficiently by physically implementing the network on neuromorphic hardware. A deep neural network structure is proposed for the sampling of arbitrary and, in general, noncommuting observables, where a phase-reweighting scheme is used in view of the complex coupling parameters. Entanglement is demonstrated through the violation of Bell inequalities.
Emergent phenomena in condensed matter physics, such as superconductivity, are rooted in the interaction of many quantum particles. These phenomena remain poorly understood in part due to the computational demands of their simulation. In recent years variational representations based on artificial neural networks, so called neural quantum states (NQS), have been shown to be efficient, ie. sub-exponentially scaling, representations. However, the computational complexity of such representations scales not only with the size of the physical system, but also with the size of the neural network. In this work, we use the analog neuromorphic BrainScaleS-2 platform to implement probabilistic representations of two particular types of quantum states. The physical nature of the neuromorphic system enforces an inherent parallelism of the compuation, rendering the emulation time independent of the used network size. We show the effectiveness of our scheme in two settings: First, we consider a hallmark test for ”quantumness” by representing a quantum state that violates the classical bounds of the Bell inequality. Second, we show that we can represent the large class of stoquastic quantum states with fidelities above 98% for moderate system sizes. This offers a novel application for spike-based neuromorphic hardware which departs from the more traditional neuroscience-inspired use cases.
Artificial neural networks were recently shown to be an efficient representation of highly entangled many-body quantum states. In practical applications, neural-network states inherit numerical schemes used in variational Monte Carlo method, most notably the use of Markov-chain Monte Carlo (MCMC) sampling to estimate quantum expectations. The local stochastic sampling in MCMC caps the potential advantages of neural networks in two ways: (i) Its intrinsic computational cost sets stringent practical limits on the width and depth of the networks, and therefore limits their expressive capacity; (ii) its difficulty in generating precise and uncorrelated samples can result in estimations of observables that are very far from their true value. Inspired by the state-of-the-art generative models used in machine learning, we propose a specialized neural-network architecture that supports efficient and exact sampling, completely circumventing the need for Markov-chain sampling. We demonstrate our approach for two-dimensional interacting spin models, showcasing the ability to obtain accurate results on larger system sizes than those currently accessible to neural-network quantum states.
The approximation of quantum states with artificial neural networks has gained a lot of attention during the last years. Meanwhile, analog neuromorphic chips, inspired by structural and dynamical properties of the biological brain, show a high energy efficiency in running artificial neural-network architectures for the profit of generative applications. This encourages employing such hardware systems as platforms for simulations of quantum systems. Here we report on the realization of a prototype using the latest spike-based BrainScaleS hardware allowing us to represent few-qubit maximally entangled quantum states with high fidelities. Bell correlations of pure and mixed two-qubit states are well captured by the analog hardware, demonstrating an important building block for simulating quantum systems with spiking neuromorphic chips.
This paper introduces a novel encoding scheme for quantum neural networks (QNNs) designed to leverage the inherent parallelism of neuromorphic quantum computing hardware and the contextual awareness of neuromorphic computation. The core idea is to develop a context-aware encoding method based on quantum state superposition and entanglement, where each quantum bit (qubit) carries contextual information related to its surrounding environment, including the states of neighboring qubits and the synaptic weights within the neuromorphic network. This contextual information is processed by the neuromorphic network to dynamically adjust the parameters of the QNN, thereby enhancing its learning efficiency and generalization capabilities, particularly in complex tasks. We demonstrate that this approach contrasts sharply with traditional QNN encoding techniques which predominantly rely on fixed parameter representations, and we argue that it represents a significant step towards overcoming performance limitations in QNNs. The proposed method offers the potential to significantly improve the learning speed and robustness of QNNs in scenarios with high data dimensionality and complex correlations, representing a new paradigm for quantum machine learning. ---
Part of the challenge for quantum many-body problems comes from the difficulty of representing large-scale quantum states, which in general requires an exponentially large number of parameters. Neural networks provide a powerful tool to represent quantum many-body states. An important open question is what characterizes the representational power of deep and shallow neural networks, which is of fundamental interest due to the popularity of deep learning methods. Here, we give a proof that, assuming a widely believed computational complexity conjecture, a deep neural network can efficiently represent most physical states, including the ground states of many-body Hamiltonians and states generated by quantum dynamics, while a shallow network representation with a restricted Boltzmann machine cannot efficiently represent some of those states.One of the challenges in studies of quantum many-body physics is finding an efficient way to record the large system wavefunctions. Here the authors present an analysis of the capabilities of recently-proposed neural network representations for storing physically accessible quantum states.
Recent research has demonstrated the usefulness of neural networks as variational ansatz functions for quantum many-body states. However, high-dimensional sampling spaces and transient autocorrelations confront these approaches with a challenging computational bottleneck. Compared to conventional neural networks, physical-model devices offer a fast, efficient and inherently parallel substrate capable of related forms of Markov chain Monte Carlo sampling. Here, we demonstrate the ability of a neuromorphic chip to represent the ground states of quantum spin models by variational energy minimization. We develop a training algorithm and apply it to the transverse field Ising model, showing good performance at moderate system sizes (N≤10N\leq 10N≤10). A systematic hyperparameter study shows that scalability to larger system sizes mainly depends on sample quality, which is limited by temporal parameter variations on the analog neuromorphic chip. Our work thus provides an important step towards harnessing the capabilities of neuromorphic hardware for tackling the curse of dimensionality in quantum many-body problems.