Elevator Codes: Concatenation for resource-efficient quantum memory under biased noise

TL;DR

This work introduces a 2D local code construction that outperforms these codes for noise biases, reducing the qubit overhead by over 50% at p_Z=10^{-3}$ and $\eta = 2 \times 10^6$ to achieve a logical error rate of $10^{-12}$.

Abstract

Biased-noise qubits, in which one type of error (e.g. $X$- and $Y$-type errors) is significantly suppressed relative to the other (e.g. $Z$-type errors), can significantly reduce the overhead of quantum error correction. Codes such as the rectangular surface code or XZZX code substantially reduce the qubit overhead under biased noise, but they still face challenges. The rectangular surface code suffers from a relatively low threshold, while the XZZX code requires twice as many physical qubits to maintain the same code distance as the surface code. In this work, we introduce a 2D local code construction that outperforms these codes for noise biases $η\ge 7\times10^{4}$, reducing the qubit overhead by over 50% at $p_Z=10^{-3}$ and $η= 2 \times 10^6$ to achieve a logical error rate of $10^{-12}$. Our construction relies on the concatenation of two classical codes. The inner codes are repetition phase-flip codes while the outer codes are high-rate bit-flip codes enabled by their implementation at the logical level, which circumvents device connectivity constraints. These results indicate that under sufficiently biased noise, it is advantageous to address phase-flip and bit-flip errors at different layers of the coding scheme. The inner code should prioritize a high threshold for phase-flip errors, while the bit-flip outer code should optimize for encoding rate efficiency. In the strong biased-noise regime, high-rate outer codes keep the overhead for correcting residual bit-flip errors comparable to that of the repetition code itself, meaningfully lower than that required by earlier approaches.

Figures (6)

• Figure 1: The qubit overhead per logical qubit required at varied noise bias rates to reach a logical error rate of $10^{-12}$ for thin surface codes chamberland2022building, thin XZZX codes bonilla2021xzzx and concatenated repetition codes (this work). The phase-flip error rate is fixed at $p_Z=10^{-3}$ while the noise-bias $\eta$ is varied. The smallest possible code of each type was selected for each noise bias level. For concatenated codes, small changes in overhead are caused by the addition of an extra ancilla inner code block to $\mathcal{C}^{\text{outer}}$. All logical error rates are reported per round of syndrome extraction, using the inner repetition code round for concatenated codes.
• Figure 2: Layout of a concatenated repetition code. $\mathcal{C}^\text{inner}$ are repetition codes and $\mathcal{C}^\text{outer}$ is a $n=5$ qubit code. The 6 blocks of separate $\mathcal{C}^\text{inner}$ codes are the input data and ancilla qubits of one block of $\mathcal{C}^\text{outer}$. Logical CNOT and SWAP gates are executed by transversal CNOT gates to implement the syndrome extraction of $\mathcal{C}^\text{outer}$ using the ancilla logical qubit.
• Figure 3: Qubit overhead as a function of the logical error rate with repetition codes guillaud2019repetition, concatenated repetition codes (this work), thin surface codes chamberland2022building and thin XZZX codes bonilla2021xzzx. The qubit overhead includes ancilla qubits and is reported per logical qubit, while the logical error rate is reported per error correction cycle, with the cycle of the inner repetition code used for concatenated codes. (a) Heavily biased error model with deep threshold phase-flip error rates $p_Z$. Concatenated codes offer lower overhead than thin surface codes and thin XZZX codes, and remain effective up to a logical error rate of about $10^{-15}$. While the repetition code has lower overhead, the logical error rates it can achieve are limited by its inability to correct bit-flip errors. (b) Intermediate error regime for medium-term biased-noise qubits. Concatenated codes have a lower overhead than thin surface and thin XZZX codes up to a logical error rate of $10^{-11}$.
• Figure 4: Parity-check matrix of the $[15,9,3]$ code.
• Figure 5: Parity-check matrix of the $[15,6,5]$ code.