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

TL;DR

This paper presents a new proof of the area law for 1D gapped quantum systems, improving previous bounds and providing a subexponential algorithm for approximating ground states and energies.

Contribution

It introduces a novel approach using a Chebyshev-based AGSP directly from the Hamiltonian, extending applicability to frustrated systems and improving the entanglement bound.

Findings

Proves an improved area law bound of $O(rac{ ext{log}^3 d}{ ext{eps}})$ for 1D gapped systems.

Shows the ground state can be approximated by a matrix product state with sublinear bond dimension.

Provides a subexponential time algorithm for ground state energy approximation.

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})}).