Imperfect measurement can degrade a quantum error correction scheme. A solution that restores fault tolerance is to add redundancy to the process of syndrome extraction. In this work, we show how to optimize this process for an arbitrary ratio of data qubit error probability to measurement error probability. The key is to design the measurements so that syndromes that correspond to different errors are separated by the maximum distance in the signal space, in close analogy to classical error correction codes. We find that the mathematical theory of 2-designs, appropriately modified, is the right tool for this. Analytical and simulation results for the bit-flip code, the 5-qubit code, and the Steane code are presented. The results show that design-based redundancy protocols show improvement in both cost and performance relative to conventional fault-tolerant error-correction schemes in situations, quite important in practice, where measurement errors are common. In the near term, the construction of a fault-tolerant logical qubit with a small number of noisy physical qubits will benefit from targeted redundancy in syndrome extraction.

不完美的测量会降低量子误差校正方案的质量。恢复容错能力的解决方案是在校正子提取过程中增加冗余。在这项工作中，我们展示了如何针对数据量子位错误概率与测量错误概率的任意比率优化此过程。关键是设计测量值，以使与不同错误相对应的校验子被信号空间中的最大距离分隔开，这与经典的纠错码非常相似。我们发现，对2设计的数学理论进行适当的修改是解决此问题的正确工具。给出了比特翻转码，5比特码和Steane码的分析和仿真结果。结果表明，与传统的容错错误纠正方案相比，基于设计的冗余协议在成本和性能方面均得到了改善，在实际情况中（测量误差很普遍）在实践中非常重要。在短期内，具有少量嘈杂物理量子位的容错逻辑量子位的构造将受益于校正子提取中的目标冗余。