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


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.


Detects a single-qubit error.


Adaptive constant-depth circuit with geometrically local gates and measurements throughout [3][4].

Transversal Gates

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


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].


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


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




A. M. Steane, “Simple quantum error-correcting codes”, Physical Review A 54, 4741 (1996) arXiv:quant-ph/9605021 DOI
D. Gottesman, “Theory of fault-tolerant quantum computation”, Physical Review A 57, 127 (1998) arXiv:quant-ph/9702029 DOI
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
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
C. N. Self, M. Benedetti, and D. Amaro, “Protecting Expressive Circuits with a Quantum Error Detection Code”, (2022) arXiv:2211.06703
R. Chao and B. W. Reichardt, “Quantum Error Correction with Only Two Extra Qubits”, Physical Review Letters 121, (2018) arXiv:1705.02329 DOI
R. Chao and B. W. Reichardt, “Fault-tolerant quantum computation with few qubits”, npj Quantum Information 4, (2018) arXiv:1705.05365 DOI
J. Preskill. Lecture notes on Quantum Computation. (1997–2020) URL
M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004) arXiv:quant-ph/0312164 DOI
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Zoo Code ID: iceberg

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
@incollection{eczoo_iceberg, title={\([[2m,2m-2,2]]\) error-detecting code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/iceberg} }
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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

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