\([[16,6,4]]\) Tesseract color code

\([[16,6,4]]\) Tesseract color code[1,2]

Description

A (self-dual CSS) 4D color code defined on a tesseract, with stabilizer generators of both types supported on each cube. A \([[16,4,2,4]]\) tesseract subsystem code can be obtained from this code by using two logical qubits as gauge qubits [3].

Transversal Gates

Global transversal \(S\) implements a logical circuit composed of \(CZ\) and \(Z\) gates [4,4,5]

Decoding

Post-selected fault-tolerant syndrome extraction [1,2].

Fault Tolerance

Post-selected fault-tolerant syndrome extraction [1,2].

Realizations

Trapped-ion devices: logical graph and GHZ states of up to 12 logical qubits constructed using three copies of the \([[16,4,2,4]]\) tesseract subsystem code, along with five rounds of post-selected fault-tolerant error correction in a device by Quantinuum [3].Neutral atom arrays: deep-circuit protocols with dozens of logical qubits and hundreds of logical teleportations by the Lukin group [6].

Cousins

\([[15, 7, 3]]\) quantum Hamming code— The \([[15,7,3]]\) quantum Hamming code can be obtained by puncturing the tesseract color code [1].
\([[8,3,2]]\) Smallest interesting color code— Applying CNOT gates to the tesseract color code disentangles it into two \([[8,3,2]]\) color codes [3].
\([[4,2,2]]\) Four-qubit code— The \([[16,4,2,4]]\) tesseract subsystem color code with particular gauge fixing can be obtained from four copies of the \([[4,2,2]]\) code [3].
Hypercube code— Stabilizer generators of both types of the tesseract color code are supported on each cube of a tesseract [1,2].

Member of code lists

Primary Hierarchy

References

[1]: N. Delfosse and B. W. Reichardt, “Short Shor-style syndrome sequences”, (2020) arXiv:2008.05051
[2]: P. Prabhu and B. W. Reichardt, “Distance-four quantum codes with combined postselection and error correction”, Physical Review A 110, (2024) arXiv:2112.03785 DOI
[3]: B. W. Reichardt et al., “Demonstration of quantum computation and error correction with a tesseract code”, (2024) arXiv:2409.04628
[4]: A. Barg, N. J. Coble, D. Hangleiter, and C. Kang, “Geometric structure and transversal logic of quantum Reed-Muller codes”, (2024) arXiv:2410.07595
[5]: N. Rengaswamy, R. Calderbank, M. Newman, and H. D. Pfister, “On Optimality of CSS Codes for Transversal T”, IEEE Journal on Selected Areas in Information Theory 1, 499 (2020) DOI
[6]: D. Bluvstein et al., “Architectural mechanisms of a universal fault-tolerant quantum computer”, (2025) arXiv:2506.20661

Page edit log

Victor V. Albert (2024-09-10) — most recent
Nolan Coble (2025-01-23)

Cite as:

“\([[16,6,4]]\) Tesseract color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/stab_16_6_4

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/color/stab_16_6_4.yml.