\([[2m,2m-2,2]]\) error-detecting code

\([[2m,2m-2,2]]\) error-detecting code[1–3]

Alternative names: Iceberg code, \([[2m,2m-2,2]]\) quantum parity code.

Description

Self-complementary CSS code for \(m\geq 2\) with generators \(\{XX\cdots X, ZZ\cdots Z\} \) acting on all \(2m\) physical qubits. The code is constructed via the CSS construction from an SPC code and a repetition code [4; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [5].

Admits a basis such that each codeword is a superposition of a computational basis state labeled by an even-weight bitstring \(b\) and a state labeled by the negation of \(b\). Its all-zero logical state is a conventional GHZ state. Removing the \(Z\)-type generator expands the number of codewords to include superpositions of odd-weight bitstrings and their negations.

All of its automorphisms lie in the Clifford group [6; Thm. 13].

Protection

Detects a single-qubit error.

Encoding

Adaptive constant-depth circuit with geometrically local gates and measurements throughout [7,8].

Transversal Gates

Transveral CNOT gates can be performed by first teleporting qubits into different code blocks [2; Sec. VII].

Gates

Logical SWAP gates can be performed fault tolerantly using an ancilla qubit [2; Sec. VII].Universal set of gates, each of which is supported on two qubits [9].Fault-tolerant Clifford Trotter circuits that are linear in \(k\) using flag qubits via a solve-and-stitch algorithm and application of a logical identity circuit [10].

Decoding

The \([[2m,2m-2,2]]\) error-detecting code [11] and its relative the code with single stabilizer \(XX\cdots X\) [12] admit continuous-time QEC against single AD errors.

Fault Tolerance

Logical SWAP gates can be performed fault tolerantly using an ancilla qubit [2; Sec. VII].Two-qubit fault-tolerant state preparation, error detection and projective measurements [13] (see also [9]).CNOT and Hadamard gates using only two extra qubits and four-qubit fault-tolerant CCZ gate [14].Fault-tolerant Clifford Trotter circuits using flag qubits [10].Weak fault tolerance: any single gate error can be detected by measuring stabilizers and utilizing extra ancillas [15].

Realizations

Trapped-ion devices: the \(m=5\) code has been realized on a 12-qubit device by Quantinuum [9].

Notes

See description of the code in Ref. [16].The code is useful for entanglement distillation [17].The code is used in a fault-tolerant implementation of the QAOA algorithm [18].

Cousins

Single parity-check (SPC) code— The \([[2m,2m-2,2]]\) error-detecting code is constructed via the CSS construction from an SPC code and its dual repetition code [4; Sec. III].
Repetition code— The \([[2m,2m-2,2]]\) error-detecting code is constructed via the CSS construction from an SPC code and its dual repetition code [4; Sec. III].
Truncated trihexagonal (4.6.12) color code— The \([[2m,2m-2,2]]\) error-detecting code for \(m=4\) is a color code defined on a single octagon of the 6.6.6 or 4.6.12 tilings.
Amplitude-damping (AD) code— The \([[2m,2m-2,2]]\) error-detecting code [11] and its relative the code with single stabilizer \(XX\cdots X\) [12] admit continuous-time QEC against single AD errors.
\([[4,2,2]]_{G}\) four group-qudit code— The four group-qudit code can be extended to the \([[2m,2m-2,2]]_{G}\) group-qudit code [19; Sec. VIII]. The latter reduces to the \([[2m,2m-2,2]]\) error-detecting code for \(G=\mathbb{Z}_2\).
Jump code— The subcode of the \([[2m,2m-2,2]]\) error-detecting code consisting of codewords labeled by weight-\(m\) bitstrings is a \(((2m,\frac{1}{2}{2m \choose m},1))_{m}\) optimal jump code [20][21; Corr. 9].
Hybrid stabilizer code— The \([[2m+1,2m+2:1,2]]\) hybrid stabilizer code [22] (extendable to modular qudits [23]) is closely related to the \([[2m,2m-2,2]]\) error-detecting code.
Trapezoid subsystem code— The trapezoid code family can be obtained from the \([[2m,2m-2,2]]\) error-detecting code by using some logical qubits as gauge qubits and imposing a two-dimensional qubit geometry [24].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Qubit Hamiltonian-based QECC Quantum

Quantum multi-dimensional parity-check (QMDPC) codeCSS QLDPC Stabilizer Qubit Hamiltonian-based Concatenated quantum Small-distance block quantum QECC Quantum

Parents

Quantum multi-dimensional parity-check (QMDPC) code

The \([[2m,2m-2,2]]\) error-detecting code is a 1D QMDPC.

Quantum maximum-distance-separable (MDS) codeQECC Quantum

The \([[2m,2m-2,2]]\) error-detecting code forms one of the two families qubit quantum MDS codes [25].

Ball color codeColor Generalized homological-product QLDPC CSS Stabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum

The \([[2m,2m-2,2]]\) error-detecting code is a ball color code [26; Sec. III.A].

Self-complementary qubit codeQubit Amplitude-damping (AD) QECC AQECC Quantum

\([[2m,2m-2,2]]\) error-detecting code

Children

\([[4,2,2]]\) Four-qubit code

The \([[2m,2m-2,2]]\) error-detecting code for \(m=2\) reduces to the \([[4,2,2]]\) code.

\([[6,4,2]]\) error-detecting code

The \([[2m,2m-2,2]]\) error-detecting code for \(m=3\) reduces to the \([[6,4,2]]\) error-detecting code.

References

[1]: A. M. Steane, “Simple quantum error-correcting codes”, Physical Review A 54, 4741 (1996) arXiv:quant-ph/9605021 DOI
[2]: D. Gottesman, “Theory of fault-tolerant quantum computation”, Physical Review A 57, 127 (1998) arXiv:quant-ph/9702029 DOI
[3]: L. Viola, E. Knill, and S. Lloyd, “Dynamical Decoupling of Open Quantum Systems”, Physical Review Letters 82, 2417 (1999) arXiv:quant-ph/9809071 DOI
[4]: N. Rengaswamy, R. Calderbank, H. D. Pfister, and S. Kadhe, “Synthesis of Logical Clifford Operators via Symplectic Geometry”, 2018 IEEE International Symposium on Information Theory (ISIT) (2018) arXiv:1803.06987 DOI
[5]: A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
[6]: E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
[7]: A. B. Watts, R. Kothari, L. Schaeffer, and A. Tal, “Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuits”, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (2019) arXiv:1906.08890 DOI
[8]: R. Verresen, N. Tantivasadakarn, and A. Vishwanath, “Efficiently preparing Schrödinger’s cat, fractons and non-Abelian topological order in quantum devices”, (2022) arXiv:2112.03061
[9]: C. N. Self, M. Benedetti, and D. Amaro, “Protecting expressive circuits with a quantum error detection code”, Nature Physics 20, 219 (2024) arXiv:2211.06703 DOI
[10]: Z. Chen and N. Rengaswamy, “Tailoring Fault-Tolerance to Quantum Algorithms”, (2024) arXiv:2404.11953
[11]: C. Ahn, H. Wiseman, and K. Jacobs, “Quantum error correction for continuously detected errors with any number of error channels per qubit”, Physical Review A 70, (2004) arXiv:quant-ph/0402067 DOI
[12]: C. Ahn, H. M. Wiseman, and G. J. Milburn, “Quantum error correction for continuously detected errors”, Physical Review A 67, (2003) arXiv:quant-ph/0302006 DOI
[13]: R. Chao and B. W. Reichardt, “Quantum Error Correction with Only Two Extra Qubits”, Physical Review Letters 121, (2018) arXiv:1705.02329 DOI
[14]: R. Chao and B. W. Reichardt, “Fault-tolerant quantum computation with few qubits”, npj Quantum Information 4, (2018) arXiv:1705.05365 DOI
[15]: C. Gerhard and T. A. Brun, “Weakly Fault-Tolerant Computation in a Quantum Error-Detecting Code”, (2024) arXiv:2408.14828
[16]: J. Preskill. Lecture notes on Quantum Computation. (1997–2020) URL
[17]: C. A. Pattison, G. Baranes, J. P. B. Ataides, M. D. Lukin, and H. Zhou, “Fast quantum interconnects via constant-rate entanglement distillation”, (2024) arXiv:2408.15936
[18]: Z. He, D. Amaro, R. Shaydulin, and M. Pistoia, “Performance of Quantum Approximate Optimization with Quantum Error Detection”, (2024) arXiv:2409.12104
[19]: P. Faist, S. Nezami, V. V. Albert, G. Salton, F. Pastawski, P. Hayden, and J. Preskill, “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI
[20]: G. Alber, Th. Beth, Ch. Charnes, A. Delgado, M. Grassl, and M. Mussinger, “Detected-jump-error-correcting quantum codes, quantum error designs, and quantum computation”, Physical Review A 68, (2003) arXiv:quant-ph/0208140 DOI
[21]: T. Beth, C. Charnes, M. Grassl, G. Alber, A. Delgado, and M. Mussinger, Designs, Codes and Cryptography 29, 51 (2003) DOI
[22]: A. Nemec and A. Klappenecker, “Infinite Families of Quantum-Classical Hybrid Codes”, (2020) arXiv:1911.12260
[23]: A. Nemec and A. Klappenecker, “Nonbinary Error-Detecting Hybrid Codes”, (2020) arXiv:2002.11075
[24]: P. Singkanipa, Z. Xia, and D. A. Lidar, “Families of \(d=2\) 2D subsystem stabilizer codes for universal Hamiltonian quantum computation with two-body interactions”, (2025) arXiv:2412.06744
[25]: M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004) arXiv:quant-ph/0312164 DOI
[26]: M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI

Page edit log

Connor Clayton (2024-03-15) — most recent
Victor V. Albert (2022-12-03)

Cite as:

“\([[2m,2m-2,2]]\) error-detecting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/iceberg

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/iceberg.yml.