## Description

A qubit code which admits a basis of codewords of the form \(|c\rangle+|\overline{c}\rangle\), where \(c\) is a bitstring and \(\overline{c}\) is its negation a.k.a. complement. Their codewords generalize the two-qubit Bell states and three-qubit GHz states and are often called (qubit) cat states or poor-man's GHz states. Such codes were originally pointed out to perform well against AD noise [2].

## Protection

Self-complementary codes automatically protect against a single \(Z\) error and lie in the \(+1\)-eigenspace of the all-\(X\) Pauli string [1]. Codes consisting of computational basis states whose bitstrings are sufficiently spaced apart correct at least one AD error [2; Thm. 2.5][3; Thm. 2].

## Parents

- Qubit code
- Amplitude-damping (AD) code — Self-complementary quantum codes consisting of computational basis states whose bitstrings are sufficiently spaced apart correct at least one AD error [2; Thm. 2.5][3; Thm. 2].

## Children

- Amplitude-damping CWS code
- Smolin-Smith-Wehner (SSW) code
- \([[2^D,D,2]]\) hypercube quantum code — A basis of hypercube quantum codewords of the form \(|c\rangle+|\overline{c}\rangle\) can be obtained via the qubit CSS codeword construction since their sole \(X\)-type stabilizer generator acts on all qubits.
- \([[2m,2m-2,2]]\) error-detecting code

## Cousin

- Binary code — A binary code is called self-complementary if, for each codeword \(c\), its negation \(\overline{c}\) is also a codeword.

## References

- [1]
- J. A. Smolin, G. Smith, and S. Wehner, “Simple Family of Nonadditive Quantum Codes”, Physical Review Letters 99, (2007) arXiv:quant-ph/0701065 DOI
- [2]
- R. Lang and P. W. Shor, “Nonadditive quantum error correcting codes adapted to the ampltitude damping channel”, (2007) arXiv:0712.2586
- [3]
- P. W. Shor et al., “High performance single-error-correcting quantum codes for amplitude damping”, (2009) arXiv:0907.5149

## Page edit log

- Victor V. Albert (2024-07-14) — most recent

## Cite as:

“Self-complementary quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/self_complementary