Adaptive syndrome measurements for Shor-style error correction

TL;DR

This paper introduces an adaptive syndrome measurement method for Shor-style fault-tolerant error correction, reducing the number of measurement rounds needed and improving error correction efficiency for stabilizer codes.

Contribution

It develops an adaptive syndrome measurement protocol that optimizes the number of rounds in Shor FTEC, applicable to any stabilizer code, and verified through simulations.

Findings

Reduces the maximum number of measurement rounds compared to traditional methods.

Preserves code distance and increases pseudothreshold in simulations.

Requires no more than the code distance number of rounds on average.

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.