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

Quantum error correction (QEC) 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 an encoded subspace, and QEC performance is governed by the errors that survive decoding, not necessarily 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, 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 be used to 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.