\([[16,6,4]]\) 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].

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

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.