Efficient and simple Gibbs state preparation of the 2D toric code via duality to classical Ising chains

TL;DR

This paper introduces polynomial-depth duality transformations that relate quantum Hamiltonians to classical ones, enabling efficient Gibbs state preparation for models like the 2D toric code through duality with classical Ising chains.

Contribution

The paper develops a framework of polynomial-depth duality transformations and applies it to efficiently prepare Gibbs states of quantum Hamiltonians via classical dual models.

Findings

The 2D toric code is polynomial-depth dual to two classical Ising chains.

Duality preserves spectral gaps and mixing times in Lindbladian dynamics.

Efficient Gibbs samplers are constructed for dual classical Hamiltonians.

Abstract

We introduce the notion of polynomial-depth duality transformations, which relates two sets of operator algebras through a conjugation by a poly-depth quantum circuit, and make use of this to construct efficient Gibbs samplers for a variety of interesting quantum Hamiltonians as they are poly-depth dual to classical Hamiltonians. This is for example the case for the 2D toric code, which is demonstrated to be poly-depth dual to two decoupled classical Ising spin chains for any system size, and we give evidence that such dualities hold for a wide class of stabilizer Hamiltonians. Additionally, we extend the above notion of duality to Lindbladians in order to show that mixing times and other quantities such as the spectral gap or the modified logarithmic Sobolev inequality are preserved under duality.