## Description

A concatenated code whose outer code is a qubit code. In other words, a qubit code that can be thought of as a concatenation of an arbitrary inner code and another qubit outer code. An inner \(C = ((n_1,K,d_1))\) and outer \(C^\prime = ((n_2,2,d_2))\) qubit code yield an \(((n_1 n_2, K, d \geq d_1d_2))\) concatenated qubit code.

Concatenating an \(((n,2,d))\) qubit code can be done recursively, with the \(r\)th level of concatenation yielding an \(((n^r,2,d^r))\) code.

## Protection

Concatenating stabilizer codes can help protect against catastrophic errors such as cosmic rays [1].

## Decoding

The effective channel for a concatenation of codes is the composition of the codes' effective channels [2].Message passing algorithm for concatenated codes can be equivalent to ML decoding [3].

## Threshold

The first methods to achieve a fault-tolerant computational threshold use concatenated qubit stabilizer codes [4–10]; see the book [11]. Such methods require constant-space and polylogarithmic time overhead, but concatenations using quantum Hamming codes improve this to quasi-polylogarithmic time [12].

## Parents

## Children

- Quantum turbo code
- Auxiliary qubit mapping (AQM) code
- Concatenated Steane code
- Hierarchical code — Hierarchical codes are concatenations of constant-rate QLDPC (outer) codes with (inner) rotated surface codes. The block length of the inner code is picked to grow logarithmically with the block length of the outer code.
- Yoked surface code — A yoked surface code is a concatenation of a QMDPC code (outer code) with a rotated surface code (inner code).

## Cousins

- Hamiltonian-based code — Concatenated stabilizer code Hamiltonians have been investigated [13].
- Gauss' law code — The Gauss' law code can be concatenated to form a stabilizer code for fault-tolerant quantum simulation of the underlying gauge theory [14,15].
- Amplitude-damping (AD) code — Concatenated quantum codes can protect against qubit AD [16].
- \(((n,1,2))\) Bravyi-Lee-Li-Yoshida PI code — The Bravyi-Lee-Li-Yoshida PI code can be concatenated to yield codes that have higher distance and that admit codewords with vanishing entanglement [17; Appx. D] (cf. [18]).
- \([[2^D,D,2]]\) hypercube quantum code — The hypercube quantum code can be concatenated with a two-qubit quantum repetition code to yield a \([[2^{D+1},D,4]]\) error-detecting code family [19]. It can also be concatenated with a distance-two \(D\)-dimensional surface code to yield a \([[2^D(2^D+1),D,4]]\) error-correcting code family that admits a transversal implementation of the logical \(C^{D-1}Z\) gate [19].
- \([[9,1,3]]\) Shor code — The Shor code is a concatenation of a three-qubit bit-flip with a three-qubit phase-flip repetition code.
- \([[4,2,2]]\) Four-qubit code — The \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) \([[4,1,2]]\) subcode is the smallest QPC, i.e., a concatenation of a two-qubit bit-flip with a two-qubit phase-flip repetition code. Concatenations of \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation schemes [20] (see also Ref. [21]). Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [12,22].' Concatenating the \([[4,2,2]]\) code with the surface code is equivalent to removing stabilizer generators from the 4.8.8 color code [23]. The \([[4,2,2]]\) code can be concatenated with two copies of the surface code to yield the 4.6.12 color code [23]. An \([[8,1,2]]\) QPC correcting a single AD error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code [24–26]. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) AD errors [27]. Recursively concatenating a \([[4,1,2]]\) subcode attains a threshold [28,29].
- Five-qubit perfect code — The recursively concatenated five-qubit code has a measurement threshold of one [30]. Code performance against general Pauli channels has been worked out [2,31].
- \([[6,2,2]]\) \(C_6\) code — Concatenations of \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation schemes [20] (see also Ref. [21]) and the Meier-Eastin-Knill (MEK) magic-state distillation protocols [32]. Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [12,22].
- \([[6,4,2]]\) error-detecting code — Concatenations of this code with itself yield the level-\(r\) \([[6^r,4^r,2^r]]\) many-hypercube code [33]. The \([[6,4,2]]\) code can be concatenated with the surface code to yield the 6.6.6 color code [23; Appx. A].
- \([[8,3,2]]\) CSS code — The \([[8,3,2]]\) code can be concatenated with a 3D surface code to yield a \([[O(d^3),3,2d]]\) code family that admits a transversal implementation of the logical \(CCZ\) gate [19].
- Layer code — Each pair of surface-code squares in a layer code are fused (or quasi-concatenated) with perpendicular surface-code squares via lattice surgery.
- 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 [34].
- Raussendorf-Bravyi-Harrington (RBH) cluster-state code — Concatenation of the RBH code with small codes such as the \([[2,1,1]]\) repetition code, \([[4,1,1,2]]\) subsystem code, or Steane code can improve thresholds [35].
- \([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code — Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [12].
- \([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code — Quantum Hamming codes can be concatened with surface codes [36].
- 3D color code — The 3D color code is equivalent to multiple decoupled copies of the 3D surface code via a local constant-depth Clifford circuit [37–39]. This process can be viewed as an ungauging [40–42,42] of certain symmetries. This mapping can also be done via code concatenation [43].
- \([[15,1,3]]\) quantum Reed-Muller code — The concatenated \([[15,1,3]]\) code has a measurement threshold less than one [30].
- Subsystem homological product code — Concatenated CSS stabilizer codes are gauge-fixed SP codes [44; Thm. 4].

## References

- [1]
- Q. Xu et al., “Distributed Quantum Error Correction for Chip-Level Catastrophic Errors”, Physical Review Letters 129, (2022) arXiv:2203.16488 DOI
- [2]
- B. Rahn, A. C. Doherty, and H. Mabuchi, “Exact performance of concatenated quantum codes”, Physical Review A 66, (2002) arXiv:quant-ph/0206061 DOI
- [3]
- D. Poulin, “Optimal and efficient decoding of concatenated quantum block codes”, Physical Review A 74, (2006) arXiv:quant-ph/0606126 DOI
- [4]
- 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
- [5]
- 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
- [6]
- D. Gottesman, “Fault-tolerant quantum computation with local gates”, Journal of Modern Optics 47, 333 (2000) arXiv:quant-ph/9903099 DOI
- [7]
- D. Aharonov and M. Ben-Or, “Fault-Tolerant Quantum Computation With Constant Error Rate”, (1999) arXiv:quant-ph/9906129
- [8]
- K. M. Svore, B. M. Terhal, and D. P. DiVincenzo, “Local fault-tolerant quantum computation”, Physical Review A 72, (2005) arXiv:quant-ph/0410047 DOI
- [9]
- P. Aliferis, D. Gottesman, and J. Preskill, “Quantum accuracy threshold for concatenated distance-3 codes”, (2005) arXiv:quant-ph/0504218
- [10]
- K. M. Svore, D. P. DiVincenzo, and B. M. Terhal, “Noise Threshold for a Fault-Tolerant Two-Dimensional Lattice Architecture”, (2006) arXiv:quant-ph/0604090
- [11]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [12]
- H. Yamasaki and M. Koashi, “Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation”, Nature Physics 20, 247 (2024) arXiv:2207.08826 DOI
- [13]
- D. Bacon, “Stability of quantum concatenated-code Hamiltonians”, Physical Review A 78, (2008) arXiv:0806.2160 DOI
- [14]
- A. Rajput, A. Roggero, and N. Wiebe, “Quantum Error Correction with Gauge Symmetries”, (2022) arXiv:2112.05186
- [15]
- L. Spagnoli, A. Roggero, and N. Wiebe, “Fault-tolerant simulation of Lattice Gauge Theories with gauge covariant codes”, (2024) arXiv:2405.19293
- [16]
- T. Jackson, M. Grassl, and B. Zeng, “Concatenated codes for amplitude damping”, 2016 IEEE International Symposium on Information Theory (ISIT) (2016) arXiv:1601.07423 DOI
- [17]
- S. Bravyi et al., “How much entanglement is needed for quantum error correction?”, (2024) arXiv:2405.01332
- [18]
- G. Gour and N. R. Wallach, “Entanglement of subspaces and error-correcting codes”, Physical Review A 76, (2007) arXiv:0704.0251 DOI
- [19]
- D. Hangleiter et al., “Fault-tolerant compiling of classically hard IQP circuits on hypercubes”, (2024) arXiv:2404.19005
- [20]
- E. Knill, “Quantum computing with realistically noisy devices”, Nature 434, 39 (2005) arXiv:quant-ph/0410199 DOI
- [21]
- J. Cho, “Fault-tolerant linear optics quantum computation by error-detecting quantum state transfer”, Physical Review A 76, (2007) arXiv:quant-ph/0612073 DOI
- [22]
- S. Yoshida, S. Tamiya, and H. Yamasaki, “Concatenate codes, save qubits”, (2024) arXiv:2402.09606
- [23]
- B. Criger and B. Terhal, “Noise thresholds for the [4,2,2]-concatenated toric code”, Quantum Information and Computation 16, 1261 (2016) arXiv:1604.04062 DOI
- [24]
- G. Alber et al., “Stabilizing Distinguishable Qubits against Spontaneous Decay by Detected-Jump Correcting Quantum Codes”, Physical Review Letters 86, 4402 (2001) arXiv:quant-ph/0103042 DOI
- [25]
- T. C. Ralph, A. J. F. Hayes, and A. Gilchrist, “Loss-Tolerant Optical Qubits”, Physical Review Letters 95, (2005) arXiv:quant-ph/0501184 DOI
- [26]
- Y. Ouyang, “Avoiding coherent errors with rotated concatenated stabilizer codes”, npj Quantum Information 7, (2021) arXiv:2010.00538 DOI
- [27]
- R. Duan et al., “Multi-error-correcting amplitude damping codes”, 2010 IEEE International Symposium on Information Theory (2010) arXiv:1001.2356 DOI
- [28]
- A. M. Stephens and Z. W. E. Evans, “Accuracy threshold for concatenated error detection in one dimension”, Physical Review A 80, (2009) arXiv:0902.2658 DOI
- [29]
- Z. W. E. Evans and A. M. Stephens, “Optimal correction of concatenated fault-tolerant quantum codes”, Quantum Information Processing 11, 1511 (2011) arXiv:0902.4506 DOI
- [30]
- D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
- [31]
- B. Rahn, A. C. Doherty, and H. Mabuchi, “Exact and Approximate Performance of Concatenated Quantum Codes”, (2001) arXiv:quant-ph/0111003
- [32]
- A. M. Meier, B. Eastin, and E. Knill, “Magic-state distillation with the four-qubit code”, (2012) arXiv:1204.4221
- [33]
- H. Goto, “Many-hypercube codes: High-rate quantum error-correcting codes for high-performance fault-tolerant quantum computing”, (2024) arXiv:2403.16054
- [34]
- J. Hu, Q. Liang, and R. Calderbank, “Divisible Codes for Quantum Computation”, (2022) arXiv:2204.13176
- [35]
- Z. Li, I. Kim, and P. Hayden, “Concatenation Schemes for Topological Fault-tolerant Quantum Error Correction”, Quantum 7, 1089 (2023) arXiv:2209.09390 DOI
- [36]
- M. Fang and D. Su, “Quantum memory based on concatenating surface codes and quantum Hamming codes”, (2024) arXiv:2407.16176
- [37]
- B. Yoshida, “Classification of quantum phases and topology of logical operators in an exactly solved model of quantum codes”, Annals of Physics 326, 15 (2011) arXiv:1007.4601 DOI
- [38]
- A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
- [39]
- A. B. Aloshious, A. N. Bhagoji, and P. K. Sarvepalli, “On the Local Equivalence of 2D Color Codes and Surface Codes with Applications”, (2018) arXiv:1804.00866
- [40]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [41]
- L. Bhardwaj, D. Gaiotto, and A. Kapustin, “State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter”, Journal of High Energy Physics 2017, (2017) arXiv:1605.01640 DOI
- [42]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
- [43]
- M. Vasmer and D. E. Browne, “Three-dimensional surface codes: Transversal gates and fault-tolerant architectures”, Physical Review A 100, (2019) arXiv:1801.04255 DOI
- [44]
- W. Zeng and L. P. Pryadko, “Minimal distances for certain quantum product codes and tensor products of chain complexes”, Physical Review A 102, (2020) arXiv:2007.12152 DOI

## Page edit log

- Victor V. Albert (2024-07-16) — most recent

## Cite as:

“Concatenated qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/qubit_concatenated