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arXiv:1906.08781 · Physical Review X

Deep Quantum Geometry of Matrices

Xizhi Han, Sean A. Hartnoll

Published Mar 23, 2020• 35 citations
arXivHTMLPDF

Abstract

We employ machine learning techniques to provide accurate variational wave functions for matrix quantum mechanics, with multiple bosonic and fermionic matrices.The variational quantum Monte Carlo method is implemented with deep generative flows to search for gauge-invariant low-energy states.The ground state (and also long-lived metastable states) of an SUðNÞ matrix quantum mechanics with three bosonic matrices, and also its supersymmetric "mini-BMN" extension, are studied as a function of coupling and N. Known semiclassical fuzzy sphere states are recovered, and the collapse of these geometries in more strongly quantum regimes is probed using the variational wave function.We then describe a factorization of the quantum mechanical Hilbert space that corresponds to a spatial partition of the emergent geometry.Under this partition, the fuzzy sphere states show a boundary-law entanglement entropy in the large N limit.

DOI
10.1103/physrevx.10.011069

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We employ machine learning techniques to provide accurate variational wave functions for matrix quantum mechanics, with multiple bosonic and fermionic matrices.The variational quantum Monte Carlo method is implemented with deep generative flows to search for gauge-invariant low-energy states.The ground state (and also long-lived metastable states) of an SUðNÞ matrix quantum mechanics with three bosonic matrices, and also its supersymmetric "mini-BMN" extension, are studied as a function of coupling and N. Known semiclassical fuzzy sphere states are recovered, and the collapse of these geometries in more strongly quantum regimes is probed using the variational wave function.We then describe a factorization of the quantum mechanical Hilbert space that corresponds to a spatial partition of the emergent geometry.Under this partition, the fuzzy sphere states show a boundary-law entanglement entropy in the large N limit.
Supersymmetric quantum gauge theories are important mathematical tools in high energy physics. As an example, supersymmetric matrix models can be used as a holographic description of quantum black holes. The wave function of such supersymmetric gauge theories is not known and it is challenging to obtain with traditional techniques. We employ a neural quantum state ansatz for the wave function of a supersymmetric matrix model and use a variational quantum Monte Carlo approach to discover the ground state of the system. We discuss the difficulty of including bosonic particles and fermionic particles, as well as gauge degrees of freedom.
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Identifying variational wave functions that efficiently parametrize the physically relevant states in the exponentially large Hilbert space is one of the key tasks towards solving the quantum many-body problem. Powerful tools in this context such as tensor network states have recently been complemented by states derived from artificial neural networks (ANNs). Here, we propose and investigate a new family of quantum states, coined generalized transfer matrix states (GTMS), which bridges between the two mentioned approaches in the framework of deep ANNs. In particular, we show by means of a constructive embedding that the class of GTMS contains generic matrix product states while at the same time being capable of capturing more long-ranged quantum correlations that go beyond the area-law entanglement properties of tensor networks. While the state amplitude of generic deep ANNs cannot be exactly evaluated, that of a GTMS network can be analytically computed using transfer matrix methods. With numerical simulations, we demonstrate how the GTMS network learns random matrix product states in a supervised learning scheme, and how augmenting the network by long-ranged couplings leads to the onset of volume-law entanglement scaling. By means of an explicit example using variational Monte Carlo, we also show that GTMS can parametrize critical quantum many-body ground states to a good accuracy. Our findings suggest that GTMS are a promising candidate for the study of critical and dynamical quantum many-body systems.