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

Journal Article (Journal Article)

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.

Full Text

Duke Authors

Cited Authors

  • Huang, S; Brown, KR

Published Date

  • August 1, 2021

Published In

Volume / Issue

  • 104 / 2

Electronic International Standard Serial Number (EISSN)

  • 2469-9934

International Standard Serial Number (ISSN)

  • 2469-9926

Digital Object Identifier (DOI)

  • 10.1103/PhysRevA.104.022429

Citation Source

  • Scopus