Adaptive syndrome measurements for Shor-style error correction

Abstract

The Shor fault-tolerant error correction (FTEC) scheme uses transversal gates and ancilla qubits prepared in the cat state in syndrome extraction circuits to prevent propagation of errors caused by gate faults. For a stabilizer code of distance $d$ that can correct up to $t=\lfloor (d-1)/2\rfloor$ errors, the traditional Shor scheme handles ancilla preparation and measurement faults by performing syndrome measurements until the syndromes are repeated $t+1$ times in a row; in the worst-case scenario, $(t+1{)}^2$ rounds of measurements are required. In this work, we improve the Shor FTEC scheme using an adaptive syndrome measurement technique. The syndrome 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 require 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 simulations of FTEC protocols with the adaptive 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 protocols with the adaptive schemes require no more than $d$ rounds on average.

24 pages, 3 figures, comments welcome. V2: minor correction. V3: adding numerical simulations and classical time complexity analysis. V4: minor correction

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• Reducing Quantum Error Correction Overhead with Versatile Flag-Sharing Syndrome Extraction Circuits
• Improved performance of the Bacon-Shor code with Steane's syndrome extraction method
• Minimizing readout-induced noise for early fault-tolerant quantum computers
• Comparison of spin-qubit architectures for quantum error-correcting codes
• Weakly Fault-Tolerant Computation in a Quantum Error-Detecting Code
• Optimized Clifford Noise Reduction: Theory, Simulations and Experiments