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

TL;DR

This 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.

Abstract

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.

Figures (6)

• Figure 1: Logical failure probability for our four strategies using the $[[7,1,3]]$ code and the decoding map of \ref{['ss:713-Decoding-Map']}, shown as a function of the physical Pauli error probability $p$ for (a) $p_{\rm DD}=0.1$, (b) $p_{\rm DD}=0.01$, (c) $p_{\rm DD} = 0.001$, with perfect recovery ($p_{\rm QEC}=0$). For sufficiently small-$p$, LDD+QEC achieves the lowest logical failure probability among the encoded strategies, in agreement with \ref{['thm:Hyb.vs.QEC-asymptotics']}.
• Figure 2: Panels (a) and (b): comparison of LDD+QEC performance for different choices of LDD generators using the $[[7,1,3]]$ code at fixed $p_{\rm DD}=0.01$. (a) Imperfect recovery $p_{\rm QEC}=0.01$. (b) Perfect recovery $p_{\rm QEC}=0$. The generating sets shown are: LDD group 2084 generated by $\langle \texttt{YXYXYXY}, \texttt{XZXZXZX}\rangle$; LDD group 665 generated by $\langle \texttt{XIIYYZZ}, \texttt{ZIIXXYY}\rangle$; LDD group 72 generated by $\langle \texttt{XXXIIII}, \texttt{ZZZIIII}\rangle$; and the standard choice $\langle \texttt{XXXXXXX}, \texttt{ZZZZZZZ}\rangle$. The identity of the best-performing LDD group depends on both $p$ and $p_{\rm QEC}$, illustrating that LDD should be co-designed with the recovery map and the expected recovery imperfection level. (c) Protocol comparison for $p_{\rm QEC}=\sqrt{p}$. Although $p_{\rm QEC}=\sqrt{p}$ is outside the perfect-recovery setting of \ref{['thm:Hyb.vs.QEC']}, the hybrid LDD+QEC protocol remains favorable in the small-$p$ regime for the parameters shown.
• Figure 3: Relative improvement $R=\log_{10}(\epsilon_{\mathrm{comp}}/\epsilon_{\mathrm{hyb}})$ for the $[[7,1,3]]$ code at fixed $p=10^{-3}$, comparing LDD+QEC against (a) DD-phys, (b) LDD-only, and (c) QEC-only. Green (red) indicates parameter regions where LDD+QEC has lower (higher) logical failure probability than the comparator. Contours mark the boundary $R=0$ for several values of $p$ (as labeled), illustrating how the advantage region evolves as $p$ decreases. In (c), LDD+QEC is dominant in the entire $(p_{\rm DD} ,p_{\rm QEC})$ plane for all values of $p \in [10^{-6},0.1]$.
• Figure 4: Logical failure probability for our protocols using a $[[13,1,3]]$ code with perfect recovery ($p_{\rm QEC}=0$) and (a) $p_{\rm DD}=0.1$, (b) $p_{\rm DD}=0.01$, (c) $p_{\rm DD} = 0.001$. For the LDD generators used here, we have $\beta>\alpha$ in the notation of \ref{['thm:Hyb.vs.QEC-asymptotics']} (no minimal-weight uncorrectable error lies in the suppressed sector), i.e., the sufficient low-$p$ condition $\beta=\alpha$ is not satisfied, and correspondingly the LDD+QEC and QEC-only curves approach one another as $p\to 0$. In all other respects the protocol ordering and overall scaling behavior is qualitatively similar to that of the Steane code shown in \ref{['fig:7_1_3_comparison_Theorem2_setting']}.
• Figure 5: Logical failure probability for a $[[13,1,3]]$ code with $p_{\rm QEC}=\sqrt{p}$, comparing physical DD, LDD-only, QEC-only, and the hybrid LDD+QEC protocol. The results are qualitatively similar to those for the Steane code [see \ref{['fig:7_1_3_LDD+QEC_comparing_LDD_generating_sets']}(c)].

Theorems & Definitions (36)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1: Hybrid LDD+QEC: infidelity under an effective Pauli noise model with DD suppression and decoder failure
• proof
• Theorem 2
• Lemma 3
• proof
• proof : Proof of \ref{['thm:Hyb.vs.QEC']}