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

Also known as Iceberg code.


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].


Detects a single-qubit error.


Adaptive constant-depth circuit with geometrically local gates and measurements throughout [5,6].

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 [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].


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


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




  • 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.


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. R. Calderbank et al., “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
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”, Nature Physics (2024) arXiv:2211.06703 DOI
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
K. Yamamoto et al., “Demonstrating Bayesian Quantum Phase Estimation with Quantum Error Detection”, (2023) arXiv:2306.16608
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
P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 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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\([[2m,2m-2,2]]\) 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/edit/main/codes/quantum/qubits/small_distance/iceberg.yml.