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
Decoding
Notes
Parent
- Tensor-network code — Encoders for a concatenated codes correspond to tree tensor networks.
Children
- Group-based QPC — A group-based QPC is a concatenation of a phase-flip group-based repetition code with a bit-flip group-based repetition code.
- Concatenated bosonic 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.
- Concatenated qubit code
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].
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 et al., “Generalized concatenated quantum codes”, Physical Review A 79, (2009) arXiv:0901.1319 DOI
- [5]
- Z. Wang et al., “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, “Clifford operations and homological codes for rotors and oscillators”, (2024) arXiv:2311.07679
- [8]
- S. Beigi et al., “Graph concatenation for quantum codes”, Journal of Mathematical Physics 52, 022201 (2011) arXiv:0910.4129 DOI
- [9]
- J. Fan et al., “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