Extracting Error Thresholds through the Framework of Approximate Quantum Error Correction Condition

TL;DR

This work addresses how to define a decoder-independent intrinsic error threshold that applies to both exact and approximate quantum error correction under realistic noise. It introduces the AQEC relative entropy $S(\Lambda+B||\Lambda)$ as the central order parameter, linking it to coherent information via $S(\Lambda+B||\Lambda)=-I_c(I/K,\mathcal{N}\circ\mathcal{E})+\log K$ and to a replica-SM formulation for stabilizer codes. The authors prove that below the intrinsic threshold one can achieve asymptotically perfect recovery (in the sense of $\lim_{n\to\infty}F_e=1$), while above threshold the recovery cannot be guaranteed for finite code-space dimension; when the code dimension grows, the threshold behavior is captured by the normalized entropy $s=S/\log K$. They demonstrate this framework on perfect stabilizer codes with Weyl noise and on imperfect state preparation toric codes, showing that nonzero $B$ generally shifts or even eliminates the intrinsic threshold, consistent with phase-transition pictures in statistical mechanics. Overall, the results provide a decoder-independent criterion for memory robustness across diverse QEC codes and noise models, with implications for toric and topological codes under realistic imperfections.

Abstract

The robustness of quantum memory against physical noises is measured by two methods: the exact and approximate quantum error correction (QEC) conditions for error recoverability, and the decoder-dependent error threshold which assesses if the logical error rate diminishes with system size. Here we unravel their relations and propose a unified framework to extract an intrinsic error threshold from the approximate QEC condition, which could upper bound other decoder-dependent error thresholds. Our proof establishes that relative entropy, effectively measuring deviations from exact QEC conditions, serves as the order parameter delineating the transition from asymptotic recoverability to unrecoverability. Consequently, we establish a unified framework for determining the error threshold across both exact and approximate QEC codes, addressing errors originating from noise channels as well as those from code space imperfections. This result sharpens our comprehension of error thresholds across diverse QEC codes and error models.

Figures (3)

• Figure 1: (a) Toric code defined on 2D periodic lattice. Physical qubits stay on the edges of the lattice. The two kinds of stabilizers are $A_{v}$ defined on each vertex and $B_{p}$ defined on each plaquette as shown in the figure. The logical Pauli $Z$ operators $Z_{l_1}$ and $Z_{l_2}$ are product of $Z$'s along non-contractible loops. Correspondingly the logical Pauli $X$ operators $X_{l_1^*}$ and $X_{l_2^*}$ are defined as $X$'s along non-contractible loops on the dual lattice. (b) A circuit model of $B_{p_0}$ measurement circuit.
• Figure 2: The phase diagram. The undecodable phase stays at any finite preparation and Pauli error rates, as well as the region $e^{-4\beta}=0$ and $p > p_{th}$. Notice that the RBIM phase transition point will be different for different replica $R$, and $p_{th}$ is obtained in $R\rightarrow 1$.
• Figure 3: The circuit that underlies the definition of coherent information and entanglement entropy. Here $\mathcal{E}$ is the encoding channel, $\mathcal{R}$ is the recovery channel, $U_{\mathcal{N}}$ comes from purifying noise channel.

Theorems & Definitions (12)

• Theorem 1
• Lemma 1
• Remark
• proof : Proof of Theorem \ref{['thm1']}
• Lemma 2
• Remark
• Theorem 2
• proof : Proof of Theorem \ref{['thm2']}
• Lemma 1
• proof