\([[2m,2m-2,2]]\) 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. 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]. Subsequent experiment performed Bayesian Quantum Phase Estimation on the \(m=3\) code [8].


See description of the code in Ref. [9].




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
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
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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
  title={\([[2m,2m-2,2]]\) error-detecting code},
  booktitle={The Error Correction Zoo},
  editor={Albert, Victor V. and Faist, Philippe},
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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.