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}
The automorphism group of the underlying \([16,5,8]\) \(\text{RM}(1,4)\) code is the affine group \(AGL(4,2)\), of order \(322\,560\) [7]. Shortening \(\text{RM}(1,4)\) on one, two, and four bits yields the \([[15,7,3]]\), \([[14,8,2]]\), and \([[12,8,2]]\) codes, respectively.
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 [8,9]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 [10]. 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 [10,11].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].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 [12].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 [2,13].
- \([[15, 7, 3]]\) 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].
- \([[15,6,3]]\) quantum Hamming subcode— 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.
- \([[16,6,4]]\) copy-cup code— The \([[16,6,4]]\) copy-cup code and the \([[16,6,4]]\) tesseract color code are the two inequivalent \([[16,6,4]]\) codes whose complete logical Clifford group is realizable by depth-one two-local circuits [11]; both are pure CSS codes whose stabilizer generators all have weight eight. The two are inequivalent in that they have different weight enumerators, and one is even while the other is doubly-even self-dual.
- \([[16,4,4]]\) 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
- 4D stabilizer codes
- Color codes
- Lattice qubit stabilizer codes
- Quantum codes
- Quantum codes with fault-tolerant gadgets
- Quantum codes with notable decoders
- Quantum codes with transversal or permutation-based gates
- Quantum Reed-Muller codes
- Qubit CSS codes
- Realized quantum codes
- Small-distance qubit stabilizer codes and friends
Primary Hierarchy
Parents
Small-distance qubit stabilizer codeQubit Stabilizer Hamiltonian-based Small-distance block quantum QECC Quantum
\([[16,6,4]]\) Tesseract color code
References
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- B. W. Reichardt et al., “Demonstration of quantum computation and error correction with a tesseract code”, (2024) arXiv:2409.04628
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- S. Burton, “qecdb.org: Quantum Error Correction Database”, URL
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- F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (Elsevier, 1977)
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- 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
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- 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
- [10]
- 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
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- V. V. Albert, “Beyond transversality: structure of Clifford circuits for CSS codes”, (2026) arXiv:2608.05688
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- D. Bluvstein et al., “A fault-tolerant neutral-atom architecture for universal quantum computation”, Nature 649, 39 (2025) arXiv:2506.20661 DOI
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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