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Constructions for measuring error syndromes in Calderbank-Shor-Steane codes between Shor and Steane methods

Publication ,  Journal Article
Huang, S; Brown, KR
Published in: Physical Review A
August 1, 2021

In another work [S. Huang and K. R. Brown, Phys. Rev. Lett. 127, 090505 (2021)10.1103/PhysRevLett.127.090505] we introduced syndrome extraction methods for Calderbank-Shor-Steane quantum error-correction codes that interpolate between the well-known Shor and Steane syndrome extraction methods. Here we provide detailed proofs of the main theorems and show that up to gate ordering there is a one-to-one correspondence between extraction gadgets and partitions of the parity check matrix. Operationally, all the circuits in our framework can be obtained by merging ancilla qubits of Shor error correction with certain rules, which enables us to design fault-tolerant syndrome extraction circuits for given specific codes. We then apply our construction to the toric code and provide a detailed analysis of the time overhead of fault tolerance. In particular, we construct a syndrome extraction family whose time overhead smoothly varies from Shor to Steane error correction. To understand the potential advantage, we consider two simplified error models: no errors on the ancilla block and uncorrelated errors on the ancilla block. We study the threshold behavior of the family for both error models and show its potential advantage for quantum architectures with long coherence times.

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Published In

Physical Review A

DOI

EISSN

2469-9934

ISSN

2469-9926

Publication Date

August 1, 2021

Volume

104

Issue

2

Related Subject Headings

  • 51 Physical sciences
  • 49 Mathematical sciences
  • 34 Chemical sciences
 

Citation

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Huang, S., & Brown, K. R. (2021). Constructions for measuring error syndromes in Calderbank-Shor-Steane codes between Shor and Steane methods. Physical Review A, 104(2). https://doi.org/10.1103/PhysRevA.104.022429
Huang, S., and K. R. Brown. “Constructions for measuring error syndromes in Calderbank-Shor-Steane codes between Shor and Steane methods.” Physical Review A 104, no. 2 (August 1, 2021). https://doi.org/10.1103/PhysRevA.104.022429.
Huang, S., and K. R. Brown. “Constructions for measuring error syndromes in Calderbank-Shor-Steane codes between Shor and Steane methods.” Physical Review A, vol. 104, no. 2, Aug. 2021. Scopus, doi:10.1103/PhysRevA.104.022429.

Published In

Physical Review A

DOI

EISSN

2469-9934

ISSN

2469-9926

Publication Date

August 1, 2021

Volume

104

Issue

2

Related Subject Headings

  • 51 Physical sciences
  • 49 Mathematical sciences
  • 34 Chemical sciences