We propose a model for quantum computing with long chains of trapped ions and we design quantum error correction schemes for this model. The main components of a quantum error correction scheme are the quantum code and a quantum circuit called the syndrome extraction circuit, which is executed to perform error correction with this code. In this work, we design syndrome extraction circuits tailored to our ion chain model, a syndrome extraction tuning protocol to optimize these circuits, and we construct new quantum codes that outperform the state-of-the-art for chains of about $50$ qubits. To establish a baseline under the ion chain model, we simulate the performance of surface codes and bivariate bicycle (BB) codes equipped with our optimized syndrome extraction circuits. Then, we propose a new variant of BB codes defined by weight-five measurements, that we refer to as BB5 codes, and we identify BB5 codes that achieve a better minimum distance than any BB codes with the same number of logical qubits and data qubits, such as $[[30, 4, 5]]$ and $[[48, 4, 7]]$ BB5 codes. For a physical error rate of $10^{-3}$, the $[[48, 4, 7]]$ BB5 code achieves a logical error rate per logical qubit of $5 \cdot 10^{-5}$, which is four times smaller than the best BB code in our baseline family. It also achieves the same logical error rate per logical qubit as the distance-7 surface code but using four times fewer physical qubits per logical qubit.

翻译：本文提出了一种基于长离子链的量子计算模型，并为此模型设计了量子纠错方案。量子纠错方案的核心组件包括量子码和称为校验子提取电路的量子电路，该电路的执行可实现基于该码的纠错。在本工作中，我们设计了针对离子链模型定制的校验子提取电路、用于优化这些电路的校验子提取调谐协议，并构建了在约$50$个量子比特的链上性能优于现有技术的新型量子码。为在离子链模型下建立基准，我们模拟了配备优化校验子提取电路的面码和双变量自行车码的性能。随后，我们提出了一种由权重为五的测量定义的双变量自行车码新变体，称为BB5码，并识别出在相同逻辑量子比特数和数据量子比特数下（例如$[[30, 4, 5]]$和$[[48, 4, 7]]$ BB5码）达到比任何双变量自行车码更优最小距离的BB5码。在物理错误率为$10^{-3}$时，$[[48, 4, 7]]$ BB5码实现的每个逻辑量子比特的逻辑错误率为$5 \cdot 10^{-5}$，比基准系列中最佳的双变量自行车码低四倍。同时，它在每个逻辑量子比特上达到与距离-7面码相同的逻辑错误率，但每个逻辑量子比特使用的物理量子比特数减少了四倍。