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Improved Thermal Area Law and Quasilinear Time Algorithm for Quantum Gibbs States

Tomotaka Kuwahara1,2,*, Álvaro M. Alhambra3,†, and Anurag Anshu4,‡

  • 1Mathematical Science Team, RIKEN Center for Advanced Intelligence Project (AIP), 1-4-1 Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan
  • 2Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS) RIKEN 2-1, Hirosawa, Wako, Saitama 351-0198, Japan
  • 3Max-Planck-Institut fur Quantenoptik, D-85748 Garching, Germany
  • 4University of California, Berkeley, California 94720, USA

  • *tomotaka.kuwahara@riken.jp
  • †alvaro.alhambra@mpq.mpg.de
  • ‡anuraganshu@berkeley.edu
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Phys. Rev. X 11, 011047 – Published 9 March, 2021

DOI: https://doi.org/10.1103/PhysRevX.11.011047

Abstract

One of the most fundamental problems in quantum many-body physics is the characterization of correlations among thermal states. Of particular relevance is the thermal area law, which justifies the tensor network approximations to thermal states with a bond dimension growing polynomially with the system size. In the regime of sufficiently low temperatures, which is crucially important for practical applications, the existing techniques do not yield optimal bounds. Here, we propose a new thermal area law that holds for generic many-body systems on lattices. We improve the temperature dependence from the original O(β) to O(β2/3) up to a logarithmic factor, thereby suggesting subballistic propagation of entanglement by imaginary-time evolution. This qualitatively differs from the real-time evolution, which usually induces linear growth of entanglement. We also prove analogous bounds for the Rényi entanglement of purification and the entanglement of formation. Our analysis is based on a polynomial approximation to the exponential function which provides a relationship between the imaginary-time evolution and random walks. Moreover, for one-dimensional (1D) systems with n spins, we prove that the Gibbs state is well approximated by a matrix product operator with a sublinear bond dimension for β=o[log(n)]. This proof allows us to rigorously establish, for the first time, a quasilinear time classical algorithm for constructing a matrix product state representation of 1D quantum Gibbs states at arbitrary temperatures of β=o[log(n)]. Our new technical ingredient is a block decomposition of the Gibbs state that bears a resemblance to the decomposition of real-time evolution given by Haah et al. [Proceedings of the 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, New York, 2018), pp. 350–360].

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Popular Summary

One of the most important challenges in the study of large ensembles of quantum particles is understanding their thermal equilibrium properties. Any advances along these lines would be a boon to many problems in fields ranging from condensed-matter physics to quantum machine learning. The difficulty is the presence of strong correlations between particles in a thermal state, which can hinder efficient simulation of the particles and their behavior. Here, we show that these correlations are weaker than previously thought, an insight that can be used to develop efficient classical simulation algorithms.

The hallmark of weak correlations in quantum many-body systems is the “area-law” behavior, which says that the correlations shared between two regions are proportional to the size of the boundary (or area) between them, as opposed to their volume. In 2008, researchers proved the first area law for thermal states, revealing a linear dependence on the inverse temperature of the system. This was thought to be the best possible behavior.

In our work, contrary to this conventional belief, we clarify that the area law scales as the 2/3 power of the inverse temperature, a bit less than linearly. This implies the presence of much weaker correlations in the thermal state and leads to significant reduction of the computational resources required to simulate quantum Gibbs states.

Our analysis thus gives fundamental new insights on the structure of complex quantum thermal states and opens up a new avenue to construct qualitatively better algorithms to simulate their equilibrium properties.

Article Text

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Phys. Rev. X 11, 011047– Published 9 March, 2021
  • Received 18 August 2020
  • Revised 16 November 2020
  • Accepted 22 December 2020
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