## Description

Also known as the iceberg code. CSS stabilizer code for \(m\geq 2\) with generators \(\{XX\cdots X, ZZ\cdots Z\} \) acting on all \(2m\) physical qubits. Admits a basis such that each codeword is a superposition of a computational basis state labeled by a bitstring \(b\) and a state labeled by the negation of \(b\). Such states generalize the two-qubit Bell states and three-qubit GHz states and are often called (qubit) cat states or poor-man's GHz states.

## Protection

Detects a single-qubit error.

## Encoding

## Transversal Gates

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

## Gates

Universal set of gates, each of which is supported on two qubits [5].

## Fault Tolerance

Two-qubit fault-tolerant state preparation, error detection and projective measurements [6] (see also [5]).CNOT and Hadamard gates using only two extra qubits and four-qubit fault-tolerant CCZ gate [7].

## Realizations

Trapped-ion devices: 12-qubit device by Quantinuum [5].

## Notes

See description of the code in Ref. [8].

## Parents

- Calderbank-Shor-Steane (CSS) stabilizer code
- Quantum maximum-distance-separable (MDS) code — The \([[2m,2m-2,2]]\) error-detecting code is one of the two qubit quantum MDS [9].
- Small-distance block quantum code

## Child

## 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]
- A. B. Watts et al., “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
- [4]
- 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
- [5]
- C. N. Self, M. Benedetti, and D. Amaro, “Protecting Expressive Circuits with a Quantum Error Detection Code”, (2022) arXiv:2211.06703
- [6]
- R. Chao and B. W. Reichardt, “Quantum Error Correction with Only Two Extra Qubits”, Physical Review Letters 121, (2018) arXiv:1705.02329 DOI
- [7]
- R. Chao and B. W. Reichardt, “Fault-tolerant quantum computation with few qubits”, npj Quantum Information 4, (2018) arXiv:1705.05365 DOI
- [8]
- J. Preskill. Lecture notes on Quantum Computation. (1997–2020) URL
- [9]
- M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004) arXiv:quant-ph/0312164 DOI

## Page edit log

- Victor V. Albert (2022-12-03) — most recent

## Cite as:

“\([[2m,2m-2,2]]\) error-detecting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/iceberg