We propose a scheme for detecting and correcting faults in any Clifford circuit. The scheme is based on the observation that the set of all possible outcome bit-strings of a Clifford circuit is a linear code, which we call the outcome code. From the outcome code we construct a corresponding stabilizer code, the spacetime code. Our construction extends the circuit-to-code construction of Bacon, Flammia, Harrow and Shi [2], revisited recently by Gottesman [16], to include intermediate and multi-qubit measurements. With this correspondence, we reduce the problem of correcting faults in a circuit to the well-studied problem of correcting errors in a stabilizer code. More precisely, a most likely error decoder for the spacetime code can be transformed into a most likely fault decoder for the circuit. We give efficient algorithms to construct the outcome and spacetime codes. We also identify conditions under which these codes are LDPC, and give an algorithm to generate low-weight checks, which can then be combined with effcient LDPC code decoders.

翻译：我们提出了一种在任意Clifford电路中检测和修正错误的方案。该方案基于以下观察：Clifford电路所有可能输出比特串的集合构成一个线性码，我们称之为输出码。从输出码出发，我们构造了相应的稳定子码——时空码。我们的构造将Bacon、Flammia、Harrow和Shi [2]（近期由Gottesman [16]重新审视）提出的电路到码构造方法扩展至包含中间测量和多量子比特测量。通过这种对应关系，我们将电路中修正错误的问题简化为稳定子码中已充分研究的纠错问题。更精确地说，时空码的最大似然错误解码器可转化为电路的最大似然故障解码器。我们给出了构造输出码和时空码的高效算法，同时确定了这些码具备LDPC性质的条件，并提出了生成低权重校验子的算法，该算法可与高效的LDPC码解码器结合使用。