Concatenated quantum code[1] 

Description

A combination of two 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 \(C^\prime = ((n_2,q_1,d_2))\) block quantum code yield an \(((n_1 n_2, K, d \geq d_1d_2))_{q_2}\) concatenated 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.

Decoding

The effective channel for a concatenation of codes is the composition of the codes' effective channels [3].

Threshold

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

Notes

Concatenated codes can achieve the Gilbert-Varshamov bound [12].See the book [2] for an introduction.

Parent

Children

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
  • 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 [13; 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 two-qubit bit-flip with a two-qubit phase-flip group-based repetition code for that group.
  • Dual-rail quantum code — The KLM protocol, one of the first protocols for fault-tolerant quantum computation, utilizes concatenations of the dual-rail with a stabilizer code [14]. Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code [15].
  • \([[9,1,3]]\) Shor code — The Shor code is a concatenation of a three-qubit bit-flip with a three-qubit phase-flip repetition code.
  • \([[4,2,2]]\) CSS 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 [16]. 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 [11,17].' Concatenating the \([[4,2,2]]\) code with the surface code is equivalent to removing stabilizer generators from the 4.8.8 color code [18]. The \([[4,2,2]]\) code can be concatenated with two copies of the surface code to yield the 4.6.12 color code [18]. An \(((8,1))\) constant-excitation code correcting a single amplitude damping error can be obtained from concatenating the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) [19] subcode with the dual-rail code [15].
  • Five-qubit perfect code — The concatenated five-qubit code has a measurement threshold of one [20]. Code performance against general Pauli channels has been worked out [3,21].
  • \([[6,2,2]]\) \(C_6\) code — Concatenations of \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation schemes [16] and the Meier-Eastin-Knill (MEK) magic-state distillation protocols [22]. 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 [11,17].
  • \([[6,4,2]]\) error-detecting code — Concatenations of this code with itself yield the level-\(r\) \([[6^r,4^r,2^r]]\) many-hypercube code [23]. The \([[6,4,2]]\) code can be concatenated with the surface code to yield the 6.6.6 color code [18; Appx. A].
  • 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 [24].
  • \([[2^r-1, 2^r-2r-1, 3]]\) 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 [11].
  • \([[15,1,3]]\) quantum Reed-Muller code — The concatenated \([[15,1,3]]\) code has a measurement threshold less than one [20].
  • 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 [25–27]. This mapping can also be done via code concatenation [28].
  • Subsystem product (SP) code — Concatenated CSS stabilizer codes are gauge-fixed SP codes [29; Thm. 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 two-group-qudit bit-flip with a two-qubit phase-flip group repetition code for \(G=\mathbb{Z}_q\).

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]
B. Rahn, A. C. Doherty, and H. Mabuchi, “Exact performance of concatenated quantum codes”, Physical Review A 66, (2002) arXiv:quant-ph/0206061 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]
H. Yamasaki and M. Koashi, “Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation”, Nature Physics 20, 247 (2024) arXiv:2207.08826 DOI
[12]
Y. Ouyang, “Concatenated Quantum Codes Can Attain the Quantum Gilbert–Varshamov Bound”, IEEE Transactions on Information Theory 60, 3117 (2014) arXiv:1004.1127 DOI
[13]
Y. Xu, Y. Wang, and V. V. Albert, “Clifford operations and homological codes for rotors and oscillators”, (2024) arXiv:2311.07679
[14]
E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics”, Nature 409, 46 (2001) DOI
[15]
Y. Ouyang, “Avoiding coherent errors with rotated concatenated stabilizer codes”, npj Quantum Information 7, (2021) arXiv:2010.00538 DOI
[16]
E. Knill, “Quantum computing with realistically noisy devices”, Nature 434, 39 (2005) arXiv:quant-ph/0410199 DOI
[17]
S. Yoshida, S. Tamiya, and H. Yamasaki, “Concatenate codes, save qubits”, (2024) arXiv:2402.09606
[18]
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
[19]
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
[20]
D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
[21]
B. Rahn, A. C. Doherty, and H. Mabuchi, “Exact and Approximate Performance of Concatenated Quantum Codes”, (2001) arXiv:quant-ph/0111003
[22]
A. M. Meier, B. Eastin, and E. Knill, “Magic-state distillation with the four-qubit code”, (2012) arXiv:1204.4221
[23]
H. Goto, “Many-hypercube codes: High-rate quantum error-correcting codes for high-performance fault-tolerant quantum computation”, (2024) arXiv:2403.16054
[24]
J. Hu, Q. Liang, and R. Calderbank, “Divisible Codes for Quantum Computation”, (2022) arXiv:2204.13176
[25]
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
[26]
A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
[27]
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
[28]
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
[29]
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
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Zoo Code ID: quantum_concatenated

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“Concatenated quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_concatenated
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@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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“Concatenated quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_concatenated

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