Bounds on Autonomous Quantum Error Correction

TL;DR

This paper derives bounds on the logical error rate for autonomous quantum memories using engineered dissipation, showing conditions for arbitrarily long coherence times and superlogarithmic error decay with code size.

Contribution

It provides theoretical bounds and conditions for the performance of autonomous quantum error correction, including a model demonstrating exponential error suppression.

Findings

Bounds on logical error rates in autonomous quantum memories

Correction rate must grow with system size for long coherence

Superlogarithmic correction rate scaling enables faster error decay

Abstract

Autonomous quantum memories are a way to passively protect quantum information using engineered dissipation that creates an ``always-on'' decoder. We analyze Markovian autonomous decoders that can be implemented with a wide range of qubit and bosonic error-correcting codes, and derive several upper bounds and a lower bound on the logical error rate in terms of correction and noise rates. These bounds suggest that, in general, there is always a correction rate, possibly size-dependent, above which autonomous memories exhibit arbitrarily long coherence times. For any given autonomous memory, size dependence of this correction rate is difficult to rule out: we point to common scenarios where autonomous decoders that stochastically implement active error correction must operate at rates that grow with code size. For codes with a threshold, we show that it is possible to achieve faster-than-polynomial decay of the logical error rate with code size by using superlogarithmic scaling of the correction rate. We illustrate our results with several examples. One example is an exactly solvable global dissipative toric code model that can achieve an effective logical error rate that decreases exponentially with the linear lattice size, provided that the recovery rate grows proportionally with the linear lattice size.