Concatenated quantum code[1]
Description
A combination of two quantum codes, an inner code \(C\) and an outer code \(C^\prime\), where the physical subspace used for the inner code consists of the logical subspace of the outer code. In other words, first one encodes in the inner code \(C^\prime\), and then one encodes each of the physical registers of \(C^\prime\) in an outer code \(C\). An inner \(C = ((n_1,K,d_1))_{q_1}\) and outer \(C^\prime = ((n_2,q_1,d_2))_{q_2}\) block quantum code yield an \(((n_1 n_2, K, d \geq d_1d_2))_{q_2}\) concatenated block quantum code [2].
Concatenating an \(((n,q,d))_q\) block quantum code can be done recursively, with the \(r\)th level of concatenation yielding an \(((n^r,q,d^r))_q\) code.
Other ways to combine quantum codes include pasting [3], and generalizations of concatenation exist [4,5].
Encoding
Standard encoding proceeds by first encoding the inner code and then encoding each physical register into the outer code.Decoding
Standard decoding proceeds by first decoding the outer code and then using the resulting data to decode the inner code.Notes
See the book [2] for an introduction.Cousins
- Concatenated code— Quantum codes can be concatenated with classical codes to yield good quantum codes [6].
- Concatenated c-q code
- Rotor GKP code— The rotor GKP code can be thought of as a concatenation of a homological rotor code and a modular-qudit GKP code [7; Fig. 3].
- \([[4,2,2]]_{G}\) four group-qudit code— The \(|\overline{g_1=1,g_2}\rangle\) \([[4,1,2]]_{G}\) subcode is the smallest group-based QPC, i.e., a concatenation of a bit-flip with a phase-flip group-based repetition code for that group.
- Codeword stabilized (CWS) code— CWS codes can be concatenated by applying generalized local complementation to their underlying graphs [8].
- Modular-qudit CWS code— Generalized concatenatenations of modular-qudit CWS codes yield a family of codes that have larger logical dimension than stabilizer codes and that asymptotically approach the modular-qudit Hamming bound [4].
- \([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code— The \([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code is a concatenation of a bit-flip with a phase-flip group repetition code for \(G=\mathbb{Z}_q\).
- EA Galois-qudit stabilizer code— Concatenated EA Galois-qudit stabilizer codes have been studied [9,10].
- Galois-qudit GRS code— Concatenations of Galois-qudit GRS codes and random stabilizer codes can achieve the quantum GV bound [11].
- Galois-qudit RS code— Recursive concatenations of quantum RS codes can be asymptotically good [12].
Primary Hierarchy
Parents
Encoders for a concatenated codes correspond to tree tensor networks.
Concatenated quantum code
Children
A group-based QPC is a concatenation of a phase-flip group-based repetition code with a bit-flip group-based repetition code.
A concatenated bosonic code is a bosonic code that can be thought of as a concatenation of a possibly non-bosonic outer code and another bosonic inner code.
References
- [1]
- E. Knill and R. Laflamme, “Concatenated Quantum Codes”, (1996) arXiv:quant-ph/9608012
- [2]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [3]
- D. Gottesman, “Pasting Quantum Codes”, (1996) arXiv:quant-ph/9607027
- [4]
- M. Grassl, P. Shor, G. Smith, J. Smolin, and B. Zeng, “Generalized concatenated quantum codes”, Physical Review A 79, (2009) arXiv:0901.1319 DOI
- [5]
- Z. Wang, K. Sun, H. Fan, and V. Vedral, “Nested Quantum Error Correction Codes”, (2009) arXiv:0909.5103
- [6]
- M. Grassl, P. W. Shor, and B. Zeng, “Generalized concatenation for quantum codes”, 2009 IEEE International Symposium on Information Theory (2009) arXiv:0905.0428 DOI
- [7]
- Y. Xu, Y. Wang, and V. V. Albert, “Multimode rotation-symmetric bosonic codes from homological rotor codes”, Physical Review A 110, (2024) arXiv:2311.07679 DOI
- [8]
- S. Beigi, I. Chuang, M. Grassl, P. Shor, and B. Zeng, “Graph concatenation for quantum codes”, Journal of Mathematical Physics 52, 022201 (2011) arXiv:0910.4129 DOI
- [9]
- J. Fan, J. Li, Y. Zhou, M.-H. Hsieh, and H. V. Poor, “Entanglement-assisted concatenated quantum codes”, Proceedings of the National Academy of Sciences 119, (2022) arXiv:2202.08084 DOI
- [10]
- T. Sidana and N. Kashyap, “Entanglement-Assisted Quantum Error-Correcting Codes over Local Frobenius Rings”, (2023) arXiv:2202.00248
- [11]
- Y. Ouyang, “Concatenated Quantum Codes Can Attain the Quantum Gilbert–Varshamov Bound”, IEEE Transactions on Information Theory 60, 3117 (2014) arXiv:1004.1127 DOI
- [12]
- Z. Li, L.-J. Xing, and X.-M. Wang, “A family of asymptotically good quantum codes based on code concatenation”, (2008) arXiv:0901.0042
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