A Perturbative Approach to Continuous-Time Quantum Error Correction

TL;DR

This paper develops a perturbative method to analyze continuous-time quantum error correction, providing quantitative insights into its dynamics and benchmarking with stabilizer codes, revealing a transient phase followed by slow information decay.

Contribution

It introduces a perturbative expansion framework for the Lindbladian describing continuous quantum error correction, enabling analytical and numerical analysis of its dynamics.

Findings

Transient phase where error correction is ineffective

Slow decay of information after initial transient

Benchmarking with 3-qubit and 5-qubit stabilizer codes

Abstract

We present a novel discussion of the continuous-time quantum error correction introduced by Paz and Zurek in 1998 [Paz and Zurek, Proc. R. Soc. A 454, 355 (1998)]. We study the general Lindbladian which describes the effects of both noise and error correction in the weak-noise (or strong-correction) regime through a perturbative expansion. We use this tool to derive quantitative aspects of the continuous-time dynamics both in general and through two illustrative examples: the 3-qubit and the 5-qubit stabilizer codes, which can be independently solved by analytical and numerical methods and then used as benchmarks for the perturbative approach. The perturbatively accessible time frame features a short initial transient in which error correction is ineffective, followed by a slow decay of the information content consistent with the known facts about discrete-time error correction in the limit of fast operations. This behavior is explained in the two case studies through a geometric description of the continuous transformation of the state space induced by the combined action of noise and error correction.