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2-designs and redundant syndrome extraction for quantum error correction

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Abstract

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 measurement 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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Authors and Affiliations

  1. Physics Department, University of Wisconsin-Madison, 1150 Univ. Ave., Madison, WI, USA

    Vickram N. Premakumar, Hele Sha, Daniel Crow & Robert Joynt

  2. Computer Sciences Department, University of Wisconsin-Madison, Madison, WI, USA

    Eric Bach

  3. Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China

    Robert Joynt

Authors
  1. Vickram N. Premakumar
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  2. Hele Sha
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  3. Daniel Crow
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  4. Eric Bach
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  5. Robert Joynt
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Correspondence to Robert Joynt.

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Premakumar, V.N., Sha, H., Crow, D. et al. 2-designs and redundant syndrome extraction for quantum error correction. Quantum Inf Process 20, 84 (2021). https://doi.org/10.1007/s11128-021-03015-1

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