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

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

Description

A (hyperbolic 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 [5].

Transversal Gates

Global transversal \(S\) implements a logical circuit composed of \(CZ\) and \(Z\) gates [6,7]Transversal Hadamard can be chosen to swap three pairs of logical qubits, allowing even-weight Hadamard-product measurements in magic-state distillation protocols [2; Sec. III].

Gates

Using this code as a hyperbolic inner code yields quartic magic-state distillation on six outputs; pipelining it with \([[4,2,2]]\) inner-code checks lowers the non-Clifford cost from 390 to 246 noisy \(T\) gates [2; Sec. I.B.2].

Decoding

Post-selected fault-tolerant syndrome extraction [3,4].

Fault Tolerance

Post-selected fault-tolerant syndrome extraction [3,4].

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 [5].Neutral atom arrays: deep circuits and 1D-cluster-state creation using 96 logical qubits and hundreds of logical teleportations by the Lukin group [8].

Cousins

\([2^m,m+1,2^{m-1}]\) First-order RM code— The tesseract color code is constructed from the \([16,5,8]\) first-order RM code via the CSS construction [2,9].
\([[15, 7, 3]]\) quantum Hamming code— The \([[15,7,3]]\) quantum Hamming code can be obtained by puncturing the tesseract color code [3].
\([[8,3,2]]\) Smallest interesting color code— Applying CNOT gates to the tesseract color code disentangles it into two \([[8,3,2]]\) color codes [5].
\([[4,2,2]]\) 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 [5].
Hypercube code— Stabilizer generators of both types of the tesseract color code are supported on each cube of a tesseract [3,4].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Qubit QLDPC codeStabilizer Hamiltonian-based QECC Quantum

Color codeCSS Stabilizer Hamiltonian-based Qubit QECC Quantum

Parents

Color code

Quantum Reed-Muller (RM) codeCSS Stabilizer Hamiltonian-based Qubit QECC Quantum

The tesseract color code is constructed from the \([16,5,8]\) first-order RM code via the CSS construction [2,9].

4D lattice stabilizer codeLattice stabilizer Stabilizer Hamiltonian-based QECC Quantum

Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum

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

References

[1]: M. B. Hastings, “Small Majorana Fermion Codes”, (2017) arXiv:1703.00612
[2]: J. Haah, M. B. Hastings, D. Poulin, and D. Wecker, “Magic state distillation with low space overhead and optimal asymptotic input count”, Quantum 1, 31 (2017) arXiv:1703.07847 DOI
[3]: N. Delfosse and B. W. Reichardt, “Short Shor-style syndrome sequences”, (2020) arXiv:2008.05051
[4]: 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
[5]: B. W. Reichardt et al., “Demonstration of quantum computation and error correction with a tesseract code”, (2024) arXiv:2409.04628
[6]: A. Barg, N. J. Coble, D. Hangleiter, and C. Kang, “Geometric Structure and Transversal Logic of Quantum Reed–Muller Codes”, IEEE Transactions on Information Theory 72, 415 (2026) arXiv:2410.07595 DOI
[7]: 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) arXiv:1910.09333 DOI
[8]: D. Bluvstein et al., “A fault-tolerant neutral-atom architecture for universal quantum computation”, Nature 649, 39 (2025) arXiv:2506.20661 DOI
[9]: A. Gong and J. M. Renes, “Computation with quantum Reed-Muller codes and their mapping onto 2D atom arrays”, (2024) arXiv:2410.23263

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.