## 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 [3; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [4].

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.

All of its automorphisms lie in the Clifford group [5; Thm. 13].

## Protection

## Encoding

## Transversal Gates

## Gates

## Fault Tolerance

## Realizations

## Notes

## 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) code — The \([[2m,2m-2,2]]\) error-detecting code forms one of the two families qubit quantum MDS codes [13].
- Ball color code — The \([[2m,2m-2,2]]\) error-detecting code is a ball color code [14; Sec. III.A].
- Self-complementary quantum 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.

## 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 [3; 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 [3; Sec. III].
- \([[4,2,2]]_{G}\) four group-qudit code — The four group-qudit code can be extended to the \([[2m,2m-2,2]]_{G}\) group-qudit code [15; 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 [16][17; Corr. 9].
- Hybrid stabilizer code — The \([[2m+1,2m+2:1,2]]\) hybrid stabilizer code [18] (extendable to modular qudits [19]) is closely related to the \([[2m,2m-2,2]]\) error-detecting code.

## References

- [1]
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- [2]
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- [3]
- N. Rengaswamy et al., “Synthesis of Logical Clifford Operators via Symplectic Geometry”, 2018 IEEE International Symposium on Information Theory (ISIT) (2018) arXiv:1803.06987 DOI
- [4]
- A. R. Calderbank et al., “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
- [5]
- E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
- [6]
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- [7]
- 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
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- R. Chao and B. W. Reichardt, “Fault-tolerant quantum computation with few qubits”, npj Quantum Information 4, (2018) arXiv:1705.05365 DOI
- [12]
- J. Preskill. Lecture notes on Quantum Computation. (1997–2020) URL
- [13]
- M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004) arXiv:quant-ph/0312164 DOI
- [14]
- M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
- [15]
- P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI
- [16]
- G. Alber et al., “Detected-jump-error-correcting quantum codes, quantum error designs, and quantum computation”, Physical Review A 68, (2003) arXiv:quant-ph/0208140 DOI
- [17]
- T. Beth et al., Designs, Codes and Cryptography 29, 51 (2003) DOI
- [18]
- A. Nemec and A. Klappenecker, “Infinite Families of Quantum-Classical Hybrid Codes”, (2020) arXiv:1911.12260
- [19]
- A. Nemec and A. Klappenecker, “Nonbinary Error-Detecting Hybrid Codes”, (2020) arXiv:2002.11075

## 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