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  • Open Access

Analyzing Many-Body Localization with a Quantum Computer

Bela Bauer1 and Chetan Nayak1,2

  • 1Station Q, Microsoft Research, Santa Barbara, California 93106, USA
  • 2Physics Department, University of California, Santa Barbara, California 93106, USA

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Phys. Rev. X 4, 041021 – Published 3 November, 2014

DOI: https://doi.org/10.1103/PhysRevX.4.041021

Abstract

Many-body localization, the persistence against electron-electron interactions of the localization of states with nonzero excitation energy density, poses a challenge to current methods of theoretical and numerical analyses. Numerical simulations have so far been limited to a small number of sites, making it difficult to obtain reliable statements about the thermodynamic limit. In this paper, we explore the ways in which a relatively small quantum computer could be leveraged to study many-body localization. We show that, in addition to studying time evolution, a quantum computer can, in polynomial time, obtain eigenstates at arbitrary energies to sufficient accuracy that localization can be observed. The limitations of quantum measurement, which preclude the possibility of directly obtaining the entanglement entropy, make it difficult to apply some of the definitions of many-body localization used in the recent literature. We discuss alternative tests of localization that can be implemented on a quantum computer.

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With recent advances in the control of quantum systems, a small quantum computer with 50–100 qubits seems likely within the next few years. A logical question is, can we already use such a small computer to perform calculations that cannot be performed on a classical computer? This question is nontrivial, as many known quantum algorithms become advantageous over their classical counterparts only in an asymptotic regime, i.e., for a very large quantum computer with thousands of qubits. We explain how quantum computers can be used to study the dynamics and eigenstates of disordered, interacting electrons. We obtain numerical estimates for how many gates need to be coherently executed by emulating the quantum simulation on a classical computer.

The phenomenon of many-body localization has recently attracted significant attention. It revolves around the question of whether Anderson localization—the famous suppression of transport in disordered electron systems discovered by P. W. Anderson in 1958—remains stable in the presence of interactions between electrons. Numerical and analytical evidence has shown that Anderson localization does indeed remain stable, and tremendous effort has been expended to understand the details of this many-body localized phase. Significant research has been based on computer simulations; however, the best classical algorithms for the simulation of quantum systems are ill suited for this problem, and the system sizes that can be simulated have thus far been very small. We propose that a quantum computer can be used to look for signatures of many-body localization. Our goal is to measure eigenstates (which we prepare using an iterative quantum phase estimation algorithm), as well as the dynamics following local quenches. We argue that these quantum simulations can yield results that are complementary to findings from classical computers.

Our results indicate that valuable insights into many-body localization can be obtained with only moderate quantum resources.

Article Text

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Phys. Rev. X 4, 041021– Published 3 November, 2014
  • Received 16 July 2014
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This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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