Self-dual Stacked Quantum Low-Density Parity-Check Codes

TL;DR

This work introduces a method for constructing self-dual qLDPC codes by stacking non-self-dual qLDPC codes, and develops double-chain bicycle codes, double-layer bivariate bicycle codes, double-layer twisted BB codes, and double-layer reflection codes, many of which exhibit favorable code parameters.

Abstract

Quantum low-density parity-check (qLDPC) codes are promising candidates for fault-tolerant quantum computation due to their high encoding rates and distances. However, implementing logical operations using qLDPC codes presents significant challenges. Previous research has demonstrated that self-dual qLDPC codes facilitate the implementation of transversal Clifford gates. Here we introduce a method for constructing self-dual qLDPC codes by stacking non-self-dual qLDPC codes. Leveraging this methodology, we develop double-chain bicycle codes, double-layer bivariate bicycle (BB) codes, double-layer twisted BB codes, and double-layer reflection codes, many of which exhibit favorable code parameters. Additionally, we conduct numerical calculations to assess the performance of these codes as quantum memory under the circuit-level noise model, revealing that the logical failure rate can be significantly reduced with high pseudo-thresholds.

Figures (3)

• Figure 1: Illustration of a double-chain bicycle code and a double-layer BB code obtained from a bicycle code with $A=I_4+S_4^3$ and $B=I_4+S_4$ and a BB code with $A=I_{12}+T_x$ and $B=I_{12}+T_y^2$, respectively. In (a) and (c), the support of an $X$ ($Z$) stabilizer generator is highlighted by filled pink (green) circles in the upper (lower) panel. In (b) and (d), the support of an $X$ (or $Z$) stabilizer generator is indicated by filled orange circles.
• Figure 2: Logical failure rate versus physical error rate under the circuit-level noise model for (a) double-chain bicycle codes (DBC), (b) double-layer BB codes (DBBC), (c) double-layer twisted BB codes (DTBBC), and (d) double-layer reflection codes (DRC). All of these codes have logical operators of odd-weight. The grey lines represent the probability of occurrence of an error on at least one physical qubit for $k$ physical qubits, where $k$ is the number of logical qubits in qubit memory.
• Figure 3: Logical failure rate with respect to physical error rate under the circuit-level noise model for (a) double-chain bicycle codes, (b) double-layer BB codes, (c) double-layer twisted BB codes, and (d) double-layer reflection codes. All of these codes have logical operators of even-weight. The grey lines have the same meaning as those in Fig. \ref{['fig2']}. The error bars hidden behind filled circles represent the standard deviation of the logical failure rate, $\sigma_{\text{LFR}} = (1/N_c) (1-P_L)^{\frac{1}{N_c}-1} \sigma_{P_L}$, where $\sigma_{P_L} = \sqrt{\frac{P_L(1-P_L)}{N_\text{sample}}}$ is the standard deviation of the logical error rate $P_L$xu2024constantbravyi2024high.