• Open Access

Matrix-Model Simulations Using Quantum Computing, Deep Learning, and Lattice Monte Carlo

Enrico Rinaldi, Xizhi Han, Mohammad Hassan, Yuan Feng, Franco Nori, Michael McGuigan, and Masanori Hanada
PRX Quantum 3, 010324 – Published 10 February 2022

Abstract

Matrix quantum mechanics plays various important roles in theoretical physics, such as a holographic description of quantum black holes, and it underpins the only practical numerical approach to the study of complex high-dimensional supergravity theories. Understanding quantum black holes and the role of entanglement in a holographic setup is of paramount importance for the realization of a quantum theory of gravity. Moreover, a complete numerical understanding of the holographic duality and the emergence of geometric space-time features from microscopic degrees of freedom could pave the way for new discoveries in quantum information science. Euclidean lattice Monte Carlo simulations are the de facto numerical tool for understanding the spectrum of large matrix models and have been used to test the holographic duality. However, they are not tailored to extract dynamical properties or even the quantum wave function of the ground state of matrix models. Quantum computing and deep learning provide potentially useful approaches to study the dynamics of matrix quantum mechanics. If successful in the context of matrix models, these rapidly improving numerical techniques could become the new Swiss army knife of quantum gravity practitioners. In this paper, we perform the first systematic survey for quantum computing and deep-learning approaches to matrix quantum mechanics, comparing them to lattice Monte Carlo simulations. These provide baseline benchmarks before addressing more complicated problems. In particular, we test the performance of each method by calculating the low-energy spectrum.

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  • Received 23 August 2021
  • Accepted 20 December 2021

DOI:https://doi.org/10.1103/PRXQuantum.3.010324

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum Information, Science & TechnologyParticles & Fields

Authors & Affiliations

Enrico Rinaldi1,2,3,*, Xizhi Han4, Mohammad Hassan5, Yuan Feng6, Franco Nori1,2,7, Michael McGuigan8, and Masanori Hanada9

  • 1Physics Department, University of Michigan, Ann Arbor, Michigan 48109, USA
  • 2Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research, RIKEN, Wako, Saitama 351-0198, Japan
  • 3Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, Wako, Saitama 351-0198, Japan
  • 4Department of Physics, Stanford University, Stanford, California 94305-4060, USA
  • 5City College of New York, New York, New York 10031, USA
  • 6Pasadena City College, Pasadena, California 91106, USA
  • 7RIKEN Center for Quantum Computing (RQC), Wako, Saitama 351-0198, Japan
  • 8Brookhaven National Laboratory, Upton, New York 11973, USA
  • 9Department of Mathematics, University of Surrey, Guildford GU2 7XH, United Kingdom

  • *erinaldi.work@gmail.com

Popular Summary

Unifying quantum mechanics and the theory of general relativity has been a long-standing goal in physics. Using the holographic duality we can describe quantum gravitational systems, such as evaporating black holes, in terms of standard quantum mechanical systems. One such example, matrix quantum mechanics, gives us valuable insights into quantum gravity, but it is hard to solve: we need to calculate the entire quantum dynamics and extract the wave function. New computational tools, machine learning and quantum algorithms are investigated to show those can help in the quest for solving matrix models and learn about quantum gravity and the role of entanglement. This new approach can lead to the study of quantum gravity in the lab or in silicon. Two matrix models are studied, which are simple enough to be solved with more traditional methods, hence they provide a suitable benchmark. However, these models possess all the features of the more complicated matrix models used to describe black holes via the holographic duality. Here it is shown that current quantum algorithms, which are used successfully on quantum computers to solve quantum chemistry problems, i.e., the variational quantum eigensolver, are also useful to find the wave function of the ground state of a bosonic and a supersymmetric matrix models. Moreover, neural networks are utilized to approximate the quantum states of matrix models using normalizing flows. This neural quantum states allow us to scale our solutions to larger problem sizes than what is currently possible on quantum computers. These results constitute an important benchmark paving the way to future research on quantum and machine learning algorithms for quantum gravity via holography.

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Vol. 3, Iss. 1 — February - April 2022

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