Adaptive syndrome measurements for Shor-style error correction

The Shor fault-tolerant error correction (FTEC) scheme uses transversal gates and ancilla qubits prepared in the cat state in syndrome extraction circuits to pre-vent propagation of errors caused by gate faults. For a stabilizer code of distance d that can correct up to t = L(d - 1)/2 RIGHT FLOOR errors, the traditional Shor scheme han-dles ancilla preparation and measurement faults by performing syndrome measure-ments until the syndromes are repeated t + 1 times in a row; in the worst-case sce-nario, (t + 1)2 rounds of measurements are required. In this work, we improve the Shor FTEC scheme using an adaptive syn-drome measurement technique. The syn-drome for error correction is determined based on information from the differences of syndromes obtained from consecutive rounds. Our protocols that satisfy the strong and the weak FTEC conditions re-quire no more than (t+3)2/4-1 rounds and (t + 3)2/4 - 2 rounds, respectively, and are applicable to any stabilizer code. Our sim-ulations of FTEC protocols with the adap-tive schemes on hexagonal color codes of small distances verify that our protocols preserve the code distance, can increase the pseudothreshold, and can decrease the average number of rounds compared to the traditional Shor scheme. We also find that for the code of distance d, our FTEC pro -to cols with the adaptive schemes require no more than d rounds on average.

Automated summary

The Shor fault-tolerant error correction (FTEC) scheme uses transversal gates and ancilla qubits in the cat state to prevent error propagation. An adaptive syndrome measurement technique is introduced to improve the scheme. Simulations show that the proposed protocols can maintain code distance, increase pseudothreshold, and reduce the average number of measurement rounds compared to the traditional Shor scheme.