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]. 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].
Parents
- Quantum Lego code
- Holographic code — A holographic code whose encoding circuit is arranged in a tree geometry reduces to a concatenated code.
Child
- Hierarchical code — Hierarchical code is a concatenation of a constant-rate QLDPC code with a rotated surface code.
Cousins
- Concatenated 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 parity code (QPC) — A QPC is a concatenation of a phase-flip repetition code with a bit-flip repetition 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.
- \([[9,1,3]]\) Shor code — Shor's 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 & code comparison”, (2022) arXiv:2209.04573
Page edit log
- Victor V. Albert (2022-02-15) — most recent
Cite as:
“Concatenated quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_concatenated