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An area law and sub-exponential algorithm for 1D systems

Itai Arad, Alexei Kitaev, Zeph Landau, Umesh Vazirani

TL;DR

A new proof for the area law for general 1D gapped systems, which exponentially improves Hastings' famous result, and establishes a new, “random-walk like”, bound on the entanglement rank of an arbitrary power of a 1D Hamiltonian.

Abstract

We give a new proof for the area law for general 1D gapped systems, which exponentially improves Hastings' famous result \cite{ref:Has07}. Specifically, we show that for a chain of d-dimensional spins, governed by a 1D local Hamiltonian with a spectral gap \eps>0, the entanglement entropy of the ground state with respect to any cut in the chain is upper bounded by $O{\frac{\log^3 d}{\eps}}$. Our approach uses the framework Arad et al to construct a Chebyshev-based AGSP (Approximate Ground Space Projection) with favorable factors. However, our construction uses the Hamiltonian directly, instead of using the Detectability lemma, which allows us to work with general (frustrated) Hamiltonians, as well as slightly improving the $1/\eps$ dependence of the bound in Arad et al. To achieve that, we establish a new, "random-walk like", bound on the entanglement rank of an arbitrary power of a 1D Hamiltonian, which might be of independent interest: \ER{H^\ell} \le (\ell d)^{O(\sqrt{\ell})}. Finally, treating d as a constant, our AGSP shows that the ground state is well approximated by a matrix product state with a sublinear bond dimension $B=e^{O(\log^{3/4}n/\eps^{1/4})}. Using this in conjunction with known dynamical programing algorithms, yields an algorithm for a 1/\poly(n) approximation of the ground energy with a subexponential running time T\le \exp(e^{O(\log^{3/4}n/\eps^{1/4})}).

An area law and sub-exponential algorithm for 1D systems

TL;DR

A new proof for the area law for general 1D gapped systems, which exponentially improves Hastings' famous result, and establishes a new, “random-walk like”, bound on the entanglement rank of an arbitrary power of a 1D Hamiltonian.

Abstract

We give a new proof for the area law for general 1D gapped systems, which exponentially improves Hastings' famous result \cite{ref:Has07}. Specifically, we show that for a chain of d-dimensional spins, governed by a 1D local Hamiltonian with a spectral gap \eps>0, the entanglement entropy of the ground state with respect to any cut in the chain is upper bounded by . Our approach uses the framework Arad et al to construct a Chebyshev-based AGSP (Approximate Ground Space Projection) with favorable factors. However, our construction uses the Hamiltonian directly, instead of using the Detectability lemma, which allows us to work with general (frustrated) Hamiltonians, as well as slightly improving the dependence of the bound in Arad et al. To achieve that, we establish a new, "random-walk like", bound on the entanglement rank of an arbitrary power of a 1D Hamiltonian, which might be of independent interest: \ER{H^\ell} \le (\ell d)^{O(\sqrt{\ell})}. Finally, treating d as a constant, our AGSP shows that the ground state is well approximated by a matrix product state with a sublinear bond dimension $B=e^{O(\log^{3/4}n/\eps^{1/4})}. Using this in conjunction with known dynamical programing algorithms, yields an algorithm for a 1/\poly(n) approximation of the ground energy with a subexponential running time T\le \exp(e^{O(\log^{3/4}n/\eps^{1/4})}).

Paper Structure

This paper contains 11 sections, 14 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background: Approximate Ground State Projectors and their consequences
  3. Overview
  4. Approximate Ground State Projector
  5. Low bond dimension MPS for frustration free 1D Hamiltonians
  6. Frustrated Case
  7. The operator $H^{(t)}$
  8. General 1D Area Law
  9. Proof of Theorem \ref{['thm:Ht']}
  10. Sub exponential algorithm for finding the ground energy of gapped 1D Hamiltonians
  11. Acknowledgments

Key Result

Lemma 2.2

If there exists an $(D, \Delta)$-AGSP with $D\cdot\Delta \leq \frac{1}{2}$, then there is a product state ${ |{\phi} \rangle }$ whose overlap with the ground state is $\mu=|{ \langle {\Gamma} | {\phi} \rangle}| \ge 1/\sqrt{2D}$.

Figures (1)

  • Figure 1: The 1D setting. We focus on a segment of $s$ particles around the cut, denoting the multiparticle Hamiltonians to the left and right of the segment by $H_L$ and $H_R$ respectively.

Theorems & Definitions (16)

  • Definition 2.1: An Approximate Ground-Space Projection (AGSP)
  • Lemma 2.2
  • Lemma 2.3
  • Corollary 2.4: Area Law
  • Definition 3.1: Truncation
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.3
  • Theorem 6.1: Robustness Theorem
  • Theorem 6.2
  • ...and 6 more