Concatenated quantum code

Description

A combination of two codes, an inner code \(C\) and an outer code \(C^\prime\), where the physical subspace used for the outer code consists of the logical subspace of the inner code. In other words, first one encodes in the outer code \(C^\prime\), and then one encodes each of the physical registers of \(C^\prime\) in an inner code \(C\).

Threshold

The first method to achieve a fault-tolerant computational threshold uses concatenated stabilizer codes [1][2][3][4].

Parent

Cousins

  • Concatenated code
  • Quantum divisible code — A fault-tolerant \(T\) gate on the five-qubit or Steane code can be obtained by concatenating with particular quantum divisible codes.
  • Quantum parity code (QPC) — A QPC is a concatenation of a phase-flip repetition code with a bit-flip repetition code.
  • Shor \([[9,1,3]]\) code — Shor's code is a concatenation of a three-qubit bit-flip with a three-qubit phase-flip repetition code.

Zoo code information

Internal code ID: quantum_concatenated

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: quantum_concatenated

Cite as:
“Concatenated quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_concatenated
BibTeX:
@incollection{eczoo_quantum_concatenated, title={Concatenated quantum code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_concatenated} }
Permanent link:
https://errorcorrectionzoo.org/c/quantum_concatenated

References

[1]
E. Knill, R. Laflamme, and W. H. Zurek, “Resilient quantum computation: error models and thresholds”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 365 (1998). DOI; quant-ph/9702058
[2]
Dorit Aharonov and Michael Ben-Or, “Fault-Tolerant Quantum Computation With Constant Error Rate”. quant-ph/9906129
[3]
J. Preskill, “Reliable quantum computers”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 385 (1998). DOI; quant-ph/9705031
[4]
Panos Aliferis, Daniel Gottesman, and John Preskill, “Quantum accuracy threshold for concatenated distance-3 codes”. quant-ph/0504218

Cite as:

“Concatenated quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_concatenated

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/properties/quantum_concatenated.yml.