Rigorous RG algorithms and area laws for low energy eigenstates in 1D

TL;DR

This paper introduces rigorous algorithms for finding low energy states in 1D quantum systems, proving area laws and providing efficient classical descriptions for degenerate and low-energy states, advancing understanding of quantum many-body complexity.

Contribution

The authors develop a new RG-based algorithm for 1D systems that efficiently finds low energy states and proves area laws for degenerate and low-energy spaces, resolving key open questions.

Findings

Polynomial time algorithm for degenerate ground spaces

$n^{O(\log n)}$ algorithm for lowest energy states

Proof of area laws and classical descriptions for these states

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 of Landau et al. gave a polynomial time algorithm to actually compute a succinct classical description for unique ground states of gapped 1D quantum systems. Despite this progress many questions remained unresolved, including whether there exist rigorous efficient algorithms when the ground space is degenerate (and poly($n$) dimensional), or for the poly($n$) lowest energy states for 1D systems, or even whether such states admit succinct classical descriptions or area laws. In this paper we give a new algorithm for finding low energy states for 1D systems, based on a rigorously justified RG type transformation. 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 for 1D systems (under a mild density condition). We note that 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 $n\times n$ matrices.