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 [1–4]. Such methods require constant-space and polylogarithmic time overhead, but concatentions using quantum Hamming codes improve this to quasi-polylogarithmic time [5].

Notes

Concatenated codes can achieve the Gilbert-Varshamov bound [6].

Parent

Children

  • Hierarchical code — Hierarchical code is a concatenation of a constant-rate QLDPC code (outer code) with a rotated surface code (inner code). The block length of the inner code is picked to grow logarithmically with the block length of the outer code.
  • Quantum parity code (QPC) — A QPC is a concatenation of a phase-flip repetition code with a bit-flip repetition code.

Cousins

  • Quantum Lego code — Concatenations of block quantum codes can be expressed as quantum Lego codes.
  • Holographic code — A holographic code whose encoding circuit is arranged in a tree geometry reduces to a concatenated code.
  • Concatenated code
  • Concatenated c-q code
  • Coherent-state constellation code — Coherent-state constellation codes consisting of points from a Gaussian quadrature rule can be concatenated with quantum polar codes to achieve the Gaussian coherent information of the thermal noise channel [7,8].
  • Quantum spherical code (QSC) — CSS codes concatenated with two-component cat codes form QSCs which have a weight-based notion of distance.
  • \(D_4\) hyper-diamond GKP code — The \(D_4\) hyper-diamond GKP code can be seen as a concatenation of a rotated square-lattice GKP code with a repetition code [9]. This is related to the fact that the four-bit repetition code yields the \(D_4\) hyper-diamond lattice code via the mod-two lattice construction.
  • GKP cluster-state code
  • GKP-stabilizer code — GKP-stabilizer oscillator-into-oscillator codes concantenated with GKP qubit-into-mode codes can outperform the more conventional concatenations of GKP codes with qubit stabilizer codes [10].
  • 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.
  • \([[9,1,3]]\) Shor code — The Shor code is a concatenation of a three-qubit bit-flip with a three-qubit phase-flip repetition code.

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) arXiv:quant-ph/9702058 DOI
[2]
D. Aharonov and M. Ben-Or, “Fault-Tolerant Quantum Computation With Constant Error Rate”, (1999) arXiv: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) arXiv:quant-ph/9705031 DOI
[4]
P. Aliferis, D. Gottesman, and J. Preskill, “Quantum accuracy threshold for concatenated distance-3 codes”, (2005) arXiv:quant-ph/0504218
[5]
H. Yamasaki and M. Koashi, “Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation”, (2022) arXiv:2207.08826
[6]
Y. Ouyang, “Concatenated Quantum Codes Can Attain the Quantum Gilbert–Varshamov Bound”, IEEE Transactions on Information Theory 60, 3117 (2014) arXiv:1004.1127 DOI
[7]
F. Lacerda, J. M. Renes, and V. B. Scholz, “Coherent-state constellations and polar codes for thermal Gaussian channels”, Physical Review A 95, (2017) arXiv:1603.05970 DOI
[8]
F. Lacerda, J. M. Renes, and V. B. Scholz, “Coherent state constellations for Bosonic Gaussian channels”, 2016 IEEE International Symposium on Information Theory (ISIT) (2016) DOI
[9]
B. Royer, S. Singh, and S. M. Girvin, “Encoding Qubits in Multimode Grid States”, PRX Quantum 3, (2022) arXiv:2201.12337 DOI
[10]
Y. Xu et al., “Qubit-Oscillator Concatenated Codes: Decoding Formalism and Code Comparison”, PRX Quantum 4, (2023) arXiv:2209.04573 DOI
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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}
}
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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.