Analytic and numerical demonstration of quantum self-correction in the 3D Cubic Code

TL;DR

This paper rigorously bounds the memory time of the 3D Cubic Code quantum memory, demonstrating conditions for self-correction and validating bounds through numerical simulations, with implications for topological stabilizer codes.

Contribution

It provides a rigorous lower bound on the memory time of the 3D Cubic Code and introduces an efficient decoding algorithm applicable to topological stabilizer codes.

Findings

Memory time scales as L^{cβ} under certain conditions

Numerical simulations confirm the tightness of the bounds

New decoding algorithm with constant error threshold

Abstract

A big open question in the quantum information theory concerns feasibility of a self-correcting quantum memory. A quantum state recorded in such memory can be stored reliably for a macroscopic time without need for active error correction if the memory is put in contact with a cold enough thermal bath. In this paper we derive a rigorous lower bound on the memory time $T_{mem}$ of the 3D Cubic Code model which was recently conjectured to have a self-correcting behavior. Assuming that dynamics of the memory system can be described by a Markovian master equation of Davies form, we prove that $T_{mem}\ge L^{c\beta}$ for some constant $c>0$, where $L$ is the lattice size and $\beta$ is the inverse temperature of the bath. However, this bound applies only if the lattice size does not exceed certain critical value $L^*\sim e^{\beta/3}$. We also report a numerical Monte Carlo simulation of the studied memory indicating that our analytic bounds on $T_{mem}$ are tight up to constant coefficients. In order to model the readout step we introduce a new decoding algorithm which might be of independent interest. Our decoder can be implemented efficiently for any topological stabilizer code and has a constant error threshold under random uncorrelated errors.