Quantum Error Correction and Dynamical Decoupling: Better Together or Apart?

TL;DR

This paper analyzes the combination of quantum error correction and dynamical decoupling, deriving formulas and conditions under which their hybrid use outperforms individual methods in protecting quantum information.

Contribution

It provides a theoretical framework with closed-form fidelity expressions for hybrid quantum memory protocols, highlighting conditions for their advantage and guiding co-design of codes and decoupling groups.

Findings

LDD+QEC outperforms QEC-only if uncorrectable errors are more prevalent in the unsuppressed sector.

A sufficient condition for hybrid advantage is LDD suppressing at least one minimum-weight uncorrectable error.

Numerical results for Steane and 13-qubit codes demonstrate parameter regimes where hybrid protocols are beneficial.

Abstract

Quantum error correction/detection (QEC/QED) and dynamical decoupling (DD) are tools for protecting quantum information. A natural goal is to combine them to outperform either approach alone. Such a benefit is not automatic: physical DD can conflict with encoded subspace, and QEC performance is governed by the errors that survive decoding, not just those DD suppresses. We analyze a hybrid memory cycle where DD is implemented logically (LDD) using normalizer elements of an $[[n,k,d]]$ stabilizer code, followed by a round of syndrome measurement and recovery (or, in the detection setting, postselection on a trivial syndrome). In an effective Pauli model with physical error probability $p$, LDD suppression factor $p_{DD}$, and recovery imperfection rate $p_{QEC}$ (or $p_{QED}$), we derive closed-form entanglement-fidelity expressions for QEC-only, QED-only, LDD-only, physical DD, and the hybrid LDD+QEC protocol. The formulas are expressed via a small set of code-dependent weight enumerator polynomials, making the role of the decoder and the LDD group explicit. For ideal recovery LDD+QEC outperforms QEC-only iff the conditional fraction of uncorrectable Pauli errors is larger in the LDD-suppressed sector than in the unsuppressed sector. In the low-noise regime, a sufficient design rule guaranteeing hybrid advantage is that LDD suppresses at least one minimum-weight uncorrectable Pauli error for the chosen recovery map. We show how stabilizer-equivalent choices of LDD generators can enforce this condition. We supplement our analysis with numerical results for the $[[7,1,3]]$ Steane code and a $[[13,1,3]]$ code, mapping regions of hybrid-protocol advantage in parameter space beyond the small-$p$ regime. Our work illustrates the need for co-design of the code, decoder, and logical decoupling group, and clarifies the conditions under which the hybrid LDD+QEC protocol is advantageous.