2-Designs and Redundant Syndrome Extraction for Quantum Error Correction. (arXiv:1907.04497v1 [quant-ph])

Imperfect measurement can degrade a quantum error correction scheme. A
solution that restores fault tolerance is to add redundancy to the process of
syndrome extraction. In this work, we show how to optimize this process for an
arbitrary ratio of data qubit error probability to measurement error
probability. The key is to design the measurements so that syndromes that
correspond to different errors are separated by the maximum distance in the
signal space, in close analogy to classical error correction codes. We find
that the mathematical theory of 2-designs, appropriately modified, is the right
tool for this. Analytical and simulation results for the bit-flip code, the
5-qubit code, and the Steane code are presented. The results show that
design-based redundancy protocols show improvement in both cost and performance
relative to conventional fault-tolerant error-correction schemes in situations,
quite important in practice, where measure errors are common. In the near term,
the construction of a fault-tolerant logical qubit with a small number of noisy
physical qubits will benefit from targeted redundancy in syndrome extraction.

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