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Communications in Mathematical Physics

, Volume 356, Issue 1, pp 65–105 | Cite as

Rigorous RG Algorithms and Area Laws for Low Energy Eigenstates in 1D

  • Itai Arad
  • Zeph Landau
  • Umesh Vazirani
  • Thomas VidickEmail author
Open Access
Article

Abstract

One of the central challenges in the study of quantum many-body systems is the complexity of simulating them on a classical computer. A recent advance (Landau et al. in Nat Phys, 2015) gave a polynomial time algorithm to compute a succinct classical description for unique ground states of gapped 1D quantum systems. Despite this progress many questions remained unsolved, including whether there exist efficient algorithms when the ground space is degenerate (and of polynomial dimension in the system size), or for the polynomially many lowest energy states, or even whether such states admit succinct classical descriptions or area laws. In this paper we give a new algorithm, based on a rigorously justified RG type transformation, for finding low energy states for 1D Hamiltonians acting on a chain of n particles. In the process we resolve some of the aforementioned open questions, including giving a polynomial time algorithm for poly(n) degenerate ground spaces and an n O(log n) algorithm for the poly(n) lowest energy states (under a mild density condition). For these classes of systems the existence of a succinct classical description and area laws were not rigorously proved before this work. The algorithms are natural and efficient, and for the case of finding unique ground states for frustration-free Hamiltonians the running time is \({\tilde{O}(nM(n))}\) , where M(n) is the time required to multiply two n × n matrices.

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Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Itai Arad
    • 1
  • Zeph Landau
    • 2
  • Umesh Vazirani
    • 2
  • Thomas Vidick
    • 3
    Email author
  1. 1.Centre for Quantum Technologies (CQT)National University of SingaporeSingaporeSingapore
  2. 2.Electrical Engineering and Computer SciencesUniversity of CaliforniaBerkeleyUSA
  3. 3.Department of Computing and Mathematical SciencesCalifornia Institute of TechnologyPasadenaUSA

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