Also known as Iceberg code.

## Description

CSS stabilizer code for \(m\geq 2\) with generators \(\{XX\cdots X, ZZ\cdots Z\} \) acting on all \(2m\) physical qubits. This is the highest-rate distance-two code when an even number of qubits is used [3].

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.

All of its automorphisms lie in the Clifford group [4; Thm. 13].

## 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 [7].

## Fault Tolerance

Two-qubit fault-tolerant state preparation, error detection and projective measurements [8] (see also [7]).CNOT and Hadamard gates using only two extra qubits and four-qubit fault-tolerant CCZ gate [9].

## Realizations

Trapped-ion devices: 12-qubit device by Quantinuum [7]. Subsequent experiment performed Bayesian Quantum Phase Estimation on the \(m=3\) code [10].

## Notes

See description of the code in Ref. [11].

## 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 is one of the two qubit quantum MDS codes [12].

## Child

## Cousins

- Four group-qudit code — The four group-qudit code can be extended to the \([[2m,2m-2,2]]_{G}\) group-qudit code [13; Sec. VIII]. The latter reduces to the \([[2m,2m-2,2]]\) error-detecting code for \(G=\mathbb{Z}_2\).
- Smolin-Smith-Wehner code — Smolin-Smith-Wehner and iceberg codewords are superpositions of particular bitstrings and their complements.

## 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. R. Calderbank et al., “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
- [4]
- E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
- [5]
- 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
- [6]
- 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
- [7]
- C. N. Self, M. Benedetti, and D. Amaro, “Protecting expressive circuits with a quantum error detection code”, Nature Physics (2024) arXiv:2211.06703 DOI
- [8]
- R. Chao and B. W. Reichardt, “Quantum Error Correction with Only Two Extra Qubits”, Physical Review Letters 121, (2018) arXiv:1705.02329 DOI
- [9]
- R. Chao and B. W. Reichardt, “Fault-tolerant quantum computation with few qubits”, npj Quantum Information 4, (2018) arXiv:1705.05365 DOI
- [10]
- K. Yamamoto et al., “Demonstrating Bayesian Quantum Phase Estimation with Quantum Error Detection”, (2023) arXiv:2306.16608
- [11]
- J. Preskill. Lecture notes on Quantum Computation. (1997–2020) URL
- [12]
- M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004) arXiv:quant-ph/0312164 DOI
- [13]
- P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 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