Layer codes as partially self-correcting quantum memories

TL;DR

Layer codes provide a concrete 3D stabilizer framework formed by layering surface-code patches according to a CSS input code, achieving BPTH-bound-optimal parameters and a polynomial energy barrier. The authors develop two decoders—a cluster decoder with a provable threshold and a concatenated MWPM-based decoder—that enable partially self-correcting memory behavior under Davies dynamics, with stronger PSCQM results for quantum Tanner layer codes and random layer codes. They show that random CSS inputs yield distance $d = \Omega(n^2/\log n)$ and energy barrier $\Delta = \Omega(n/\log n)$, with memory times scaling as $t^*_{\mathrm{mem}} = \exp(\exp(\Omega(\beta)))$ up to a cutoff $n^* = \exp(\beta/2)$, and confirm these trends via numerical simulations. Collectively, the work argues that partial self-correction is more common than previously thought when a diverging energy barrier is present and charts a practical path to PSCQM through random layer codes and efficient decoders.

Abstract

We investigate layer codes, a family of three-dimensional stabilizer codes that can achieve optimal scaling of code parameters and a polynomial energy barrier, as candidates for self-correcting quantum memories. First, we introduce two decoding algorithms for layer codes with provable guarantees for local stochastic and adversarial noise, respectively. We then prove that layer codes constitute partially self-correcting quantum memories which outperform previously analyzed models such as the cubic code and the welded solid code. Notably, we argue that partial self-correction without the requirement of efficient decoding is more common than expected, as it arises solely from a diverging energy barrier. This draws a sharp distinction between partially self-correcting systems and partially self-correcting memories. Another novel aspect of our work is an analysis of layer codes constructed from random Calderbank-Shor-Steane codes. We show that these random layer codes have optimal scaling (up to logarithmic corrections) of code parameters and a polynomial energy barrier. Finally, we present numerical studies of their memory times and report behavior consistent with partial self-correction.

Figures (13)

• Figure 1: A layer code for $H_X = H_Z = (1,1,1,1)$. Gray, red and blue layers of the surface code correspond respectively to qubits, $X$-checks and $Z$-checks. Upon crossing line defects (thick lines), excitations may split, which is determined by the fusion rules; see Appendix \ref{['sec:appendix_a']}
• Figure 2: Decoding algorithms for layer codes. Red dots represent initial excitations. (a) The cluster decoder gradually grows and merges clusters of excitations (purple shapes), and removes them if they are neutral (dashed oval); other clusters are not neutral. (b) The concatenated decoder sequentially matches excitations in the $Z$-, $Q$- and $X$-layers. At different stages 1--4, the decoder finds Pauli operators (purple lines); new excitations may also be created (pink dots labeled 1 and 3).
• Figure 3: (a) Numerical estimates $\langle t_\text{fail}\rangle$ of the memory time $t_\text{mem}$ for random layer codes and the cluster decoder as a function of the input code size $n$ for various inverse temperatures $\beta$, with error bars representing the standard error of the mean (SEM). For each $\beta$, we find the maximum memory time $t^*_\text{mem}$ at the cutoff size $n^*$ (marked by $\star$). For $n\le n^*$, we fit the data points with a numerical ansatz $\log\langle t_\text{fail}\rangle \approx a \log n + b$, where $a$ and $b$ are fitting parameters. In (b) and (c), we analyze the scaling of $n^*$ and $t^*_\text{mem}$ as a function of $\beta$, respectively, which we fit with $n^* \approx \exp(0.448\beta-0.562)$ and $t_\text{mem}^* \approx \exp(0.695\beta^2-7.11\beta+26.1)$.
• Figure 4: (a) Fusion rules for $e$ and $m$ excitations at $\bullet$- and $\oplus$-junctions. (b) Fusion rules at defect lines of layer codes defined in terms of $\bullet$- and $\oplus$-junctions.
• Figure 5: (a) A $Z$ logical operator of a layer code, illustrated as the purple strings. (b) A top-down view of the $e$-configuration at a slab boundary right above the $X$-layer. (c) A quasi-concatenated representative of the logical operator in (a). (d) The corresponding $e$-configuration, which is boundary-equivalent to the one in (b).

Theorems & Definitions (91)

• Theorem 2.1
• Theorem 3.1
• Theorem 4.1
• proof
• Theorem 4.2: informal
• Definition A.1: Energy Barrier
• Theorem A.2: Layer Codes williamson2023layer
• Remark A.3
• Definition A.4
• Lemma A.5: Lemma 3 of Supplementary information of Ref. williamson2023layer