论文标题
同构逻辑测量
Homomorphic Logical Measurements
论文作者
论文摘要
Shor和Steane Ancilla是两种易于故障逻辑测量的众所周知的方法，这些方法在小型代码及其串联上成功。然而，在大量子低密度 - 比较检查（LDPC）代码上，Shor和Steane测量值分别具有不切实际的时间和空间开销。在这项工作中，我们通过将Shor和Steane测量值统一为单个框架（称为同构测量值）来扩展Ancilla代码的选择。对于具有适当的Ancilla代码的任何Calderbank-s-s-s-s-s-s-s-s-css代码，都可以避免重复测量或复杂的Ancilla状态准备程序，例如蒸馏，这克服了Shor和Steane方法的困难。例如，我们利用覆盖空间的理论来构建针对任意$ x $ - 或$ z $ -Type逻辑的Pauli操作员的同态测量协议，包括曲面代码，包括感谢您的表面代码和夸张的表面代码。常规的表面代码解码器，例如最小重量的完美匹配，可以直接应用于我们的构造。
Shor and Steane ancilla are two well-known methods for fault-tolerant logical measurements, which are successful on small codes and their concatenations. On large quantum low-density-parity-check (LDPC) codes, however, Shor and Steane measurements have impractical time and space overhead respectively. In this work, we widen the choice of ancilla codes by unifying Shor and Steane measurements into a single framework, called homomorphic measurements. For any Calderbank-Shor-Steane (CSS) code with the appropriate ancilla code, one can avoid repetitive measurements or complicated ancilla state preparation procedures such as distillation, which overcomes the difficulties of both Shor and Steane methods. As an example, we utilize the theory of covering spaces to construct homomorphic measurement protocols for arbitrary $X$- or $Z$-type logical Pauli operators on surface codes in general, including the toric code and hyperbolic surface codes. Conventional surface code decoders, such as minimum-weight perfect matching, can be directly applied to our constructions.
