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Neuro Inspired Computational Elements Conference (NICE)

Quantum many-body states: A novel neuromorphic application

Andreas Baumbach, Robert Klassert, Stefanie Czischek, Martin Gärttner, Mihai Alexandru Petrovici

Published Mar 28, 2022• 0 citations

Abstract

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.

DOI: 10.1145/3517343.3517379

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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.

Quantum neuromorphic computing physically implements neural networks in brain-inspired quantum hardware to speed up their computation. In this perspective article, we show that this emerging paradigm could make the best use of the existing and near future intermediate size quantum computers. Some approaches are based on parametrized quantum circuits and use neural network-inspired algorithms to train them. Other approaches, closer to classical neuromorphic computing, take advantage of the physical properties of quantum oscillator assemblies to mimic neurons and synapses to compute. We discuss the different implementations of quantum neuromorphic networks with digital and analog circuits, highlight their respective advantages, and review exciting recent experimental results.

Neural quantum states provide flexible variational representations of quantum many-body wave functions by combining neural-network parametrizations with Monte Carlo sampling. In this perspective, we review recent advances in their application to condensed-matter systems, focusing on frustrated quantum magnets, interacting lattice fermions, and non-equilibrium dynamics. We discuss the architectures, symmetry constraints, optimization methods, and sampling strategies underlying state-of-the-art calculations, and summarize practical guidelines for reliable simulations. We also examine the principal remaining challenges, including learning non-trivial sign and phase structures, controlling variational bias, enforcing physical symmetries, scaling optimization to large networks, and achieving stable real-time evolution. Finally, we outline promising directions in which neural quantum states may extend the reach of classical simulations of strongly correlated quantum matter.

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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\leq 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.

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.