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

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

Alternative Names: \([[16,6,4]]\) hypercube code, \([[16,6,4]]\) 4D color code.

Description

A (hyperbolic self-dual CSS) doubly-even 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].

A stabilizer tableau for the code is [6; ID 67d2cf9965e067195651cfbe] \begin{align} \begin{smallmatrix} X & X & X & X & X & X & X & X & X & X & X & X & X & X & X & X \\ I & X & I & X & I & X & I & X & I & X & I & X & I & X & I & X \\ I & I & X & X & I & I & X & X & I & I & X & X & I & I & X & X \\ I & I & I & I & X & X & X & X & I & I & I & I & X & X & X & X \\ I & I & I & I & I & I & I & I & X & X & X & X & X & X & X & X \\ Z & Z & Z & Z & Z & Z & Z & Z & Z & Z & Z & Z & Z & Z & Z & Z \\ I & Z & I & Z & I & Z & I & Z & I & Z & I & Z & I & Z & I & Z \\ I & I & Z & Z & I & I & Z & Z & I & I & Z & Z & I & I & Z & Z \\ I & I & I & I & Z & Z & Z & Z & I & I & I & I & Z & Z & Z & Z \\ I & I & I & I & I & I & I & I & Z & Z & Z & Z & Z & Z & Z & Z \end{smallmatrix}~. \tag*{(1)}\end{align}

Transversal and Permutation-Based Gates

The logical classes of the tesseract color code are the \(63\) nontrivial cosets of \(\text{RM}(1,4)\) in \(\text{RM}(2,4)\), which split into \(35\) classes of minimum weight four, indexed by the rank-two alternating forms on \(\mathbb{F}_2^4\), and \(28\) classes of minimum weight six, indexed by the nondegenerate ones.Global transversal \(S\) implements a logical circuit composed of \(CZ\) and \(Z\) gates [7,8]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].All logical Clifford gates can be realized as two-fold transversal gates, i.e., by depth-one circuits of two-local code-preserving physical gates [9]. Twelve of the fifteen logical-qubit pairs, which can be associated with the edges of an octahedron, carry an addressable \(\overline{CZ}\) and its \(X\)-type dual, each realized by a single such circuit of four physical \(CZ\) gates and eight single-qubit gates [9,10].

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 [11].

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 \(\text{RM}(1,4)\) code via the CSS construction [2,12].
\([[15, 7, 3]]\) 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].
\([[15,6,3]]\) gauge-fixed quantum Hamming code— The \([[15,6,3]]\) code is obtained from the tesseract color code by removing one qubit, which punctures the underlying first-order RM code on the \(X\) side and shortens it to the simplex code on the \(Z\) side.
\([[16,4,4]]\) biplane code— The \([[16,4,4]]\) biplane code is obtained from the tesseract color code by adding one \(X\)-type and one \(Z\)-type stabilizer generator supported on an elliptic quadric of \(\mathbb{F}_2^4\), reducing the number of logical qubits from six to four. The promoted generators are a minimum-weight representative of one of the \(28\) logical classes of the tesseract code, which is why the resulting code remains pure.
\([[16,4,4]]\) twisted color code— The \([[16,4,4]]\) code is obtained from the tesseract color code by promoting one \(X\)-type and one \(Z\)-type weight-four logical operator to stabilizers [3,4], equivalently by fixing the weight-four gauge operators of the \([[16,4,2,4]]\) tesseract subsystem code [5].

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

The tesseract color code is a 4D color code defined on a tesseract [1–4].

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 [2,12].

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

The tesseract color code is a 4D color code defined on a tesseract [1–4].

Self-dual CSS codeCSS Stabilizer Hamiltonian-based Qubit 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]: S. Burton, “qecdb.org: Quantum Error Correction Database”, URL
[7]: 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
[8]: 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
[9]: T. Tansuwannont, T. Chan, and R. Takagi, “Construction of the full logical Clifford group for high-rate quantum Reed-Muller codes using only transversal and fold-transversal gates”, (2026) arXiv:2602.09788
[10]: V. V. Albert, “Beyond transversality: structure of Clifford circuits for CSS codes”, (2026) arXiv:2608.05688
[11]: D. Bluvstein et al., “A fault-tolerant neutral-atom architecture for universal quantum computation”, Nature 649, 39 (2025) arXiv:2506.20661 DOI
[12]: 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 (2026-06-08) — most recent
Victor V. Albert (2024-09-10)
Nolan Coble (2025-01-23)

Cite as:

“\([[16,6,4]]\) Tesseract color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_16_6_4, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/16/stab_16_6_4.yml.