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

Sergey Bravyi; Jeongwan Haah

doi:10.1103/PhysRevLett.111.200501

paper

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

arXiv:1112.3252 · doi:10.1103/PhysRevLett.111.200501

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 $T_{m e m}$ 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 $T_{m e m} \geq L^{c β}$ for some constant $c > 0$ , where $L$ is the lattice size and $β$ is the inverse temperature of the bath. However, this bound applies only if the lattice size does not exceed certain critical value $L^{*} \sim e^{β /3}$ . We also report a numerical Monte Carlo simulation of the studied memory indicating that our analytic bounds on $T_{m e m}$ 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.

40 pages, 6 figures

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

Abstract

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