# Deep Quantum Geometry of Matrices

## TL;DR

This paper uses machine learning to find accurate wavefunctions for matrix quantum mechanics, revealing how emergent geometries like fuzzy spheres behave quantum mechanically and their entanglement properties.

## Contribution

It introduces a variational quantum Monte Carlo approach with deep generative flows to study gauge invariant states in matrix quantum mechanics, including supersymmetric extensions.

## Key findings

- Recovered semiclassical fuzzy sphere states
- Probed collapse of geometries in quantum regimes
- Observed boundary-law entanglement entropy in large N limit

## Abstract

We employ machine learning techniques to provide accurate variational wavefunctions for matrix quantum mechanics, with multiple bosonic and fermionic matrices. Variational quantum Monte Carlo 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 $\mathrm{SU}(N)$ matrix quantum mechanics with three bosonic matrices, as well as 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 wavefunction. 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.