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\([[8,3,2]]\) Smallest interesting color code[1,2]

Description

Smallest 3D color code whose physical qubits lie on vertices of a cube and which admits a (weakly) transversal \(CCZ\) gate.

A stabilizer tableau for the code is given by [3; ID 4882] \begin{align} \begin{array}{cccccccc} Z & Z & I & I & I & I & Z & Z \\ Z & I & Z & Z & I & I & Z & I \\ I & I & Z & I & Z & I & Z & Z \\ Z & I & Z & I & I & Z & I & Z \\ X & X & X & X & X & X & X & X \end{array}~. \tag*{(1)}\end{align}

In encoded IQP sampling, the final measurement outcomes determine both the logical sample and stabilizer checks, enabling end-of-circuit error detection or postselected decoding directly from the classical samples [4].

Transversal Gates

\(CZ\) gates between any two logical qubits [5] and (weakly) transversal \(CCZ\) gate [1,2,5].

Gates

\(CCZ\) gate can be distilled in a fault-tolerant manner [6].Fault-tolerant and teleportation-free logical Hadamard [7].

Fault Tolerance

\(CCZ\) gate can be distilled in a fault-tolerant manner [6].Fault-tolerant and teleportation-free logical Hadamard [7].Universal weakly fault-tolerant computation via code switching between this and another \([[8,3,2]]\) CSS code in a postselected error-detecting regime [8].Fault-tolerant architecture [9].For hIQP sampling with decoding only in the final measurement round, error-detected \([[8,3,2]]\) circuits outperform the \([[16,3,4]]\) and \([[15,1,3]]\) comparison circuits studied in Ref. [4] under its two-qubit-gate-noise model.

Realizations

Trapped ions: one-qubit addition algorithm implemented fault-tolerantly on the Quantinuum H1-1 device [10]. Trapped-ion processor by AQT: measurement-free universal fault-tolerant logical operations and a Grover-search demonstration [11].Superconducting circuits: fault-tolerant \(CCZ\) gate performed on IBM and IonQ devices [12].Neutral atom arrays: Lukin group [13]. 48 logical qubits, 228 logical two-qubit gates, 48 logical \(CCZ\) gates, and error detection performed in 16 blocks. Circuit outcomes were sampled and cross-entropy (XEB) was calculated to verify quantumness. Logical entanglement entropy was measured [13].

Cousins

  • \([8,4,4]\) extended Hamming code— The \([[8,3,2]]\) hypercube code \(H_X\) check matrix is the parity-check matrix of the \([8,4,4]\) extended Hamming code, while its \(H_Z\) matrix is that of the SPC code.
  • \([n,n-1,2]\) Single parity-check (SPC) code— The \([[8,3,2]]\) hypercube code \(H_X\) check matrix is the parity-check matrix of the \([8,4,4]\) extended Hamming code, while its \(H_Z\) matrix is that of the SPC code.
  • XP stabilizer code— As the \(D=3\) member of the hypercube-code family, the \([[8,3,2]]\) code can be viewed as an XP stabilizer code with precision \(N=8\) [14; Exam. 6.10].
  • \([[15,1,3]]\) quantum RM code— The \([[8,3,2]]\) code can be obtained from a subset of physical qubits of the \([[15,1,3]]\) code [15].
  • 3D surface code— Three cyclically rotated copies of the 3D surface/toric code admit a logical \(CCZ\) gate via transversal physical \(CCZ\) gates, and concatenating each such qubit triple with an \([[8,3,2]]\) block yields a 3D toric/color family with parameters \([[8n,3,2d]]\); its smallest member has parameters \([[72,3,4]]\) [4].
  • Concatenated qubit code— Concatenating \([[8,3,2]]\) blocks with triples of qubits drawn from three cyclically rotated 3D surface/toric codes yields a 3D toric/color family with parameters \([[8n,3,2d]]\) and transversal logical \(CCZ\) implemented by physical \(T\) gates on the inner \([[8,3,2]]\) blocks [4].
  • \([[10,1,2]]\) Vasmer-Kubica code— The \([[10,1,2]]\) code is obtained by morphing the \([[15,1,3]]\) code on a region whose child code is the \([[8,3,2]]\) smallest interesting color code [15].
  • \([[12,2,2]]\) CSS code— The \([[12,2,2]]\) CSS code can be obtained by joining two copies of the \([[8,3,2]]\) code at a common face [16].
  • \([[16,6,4]]\) Tesseract color code— Applying CNOT gates to the tesseract color code disentangles it into two \([[8,3,2]]\) color codes [17].
  • \([[7,3,2]]\) punctured hypercube code— The \([[7,3,2]]\) code is obtained by puncturing one qubit from the \([[8,3,2]]\) hypercube quantum code [18].

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
3D color codeCSS Generalized homological-product Lattice stabilizer Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum
Parents
3D color code
The \([[8,3,2]]\) code is the smallest non-trivial 3D color code.
\([[2^D,D,2]]\) hypercube quantum codeColor Quantum RM code QLDPC Qubit CSS Stabilizer Hamiltonian-based Self-complementary qubit Amplitude-damping (AD) Small-distance block quantum QECC AQECC Quantum
The \([[8,3,2]]\) code is a hypercube code for \(D=3\).
\([[6k+2,3k,2]]\) Campbell-Howard codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
The \([[8,3,2]]\) code is the \(k=1\) member of the \([[6k+2,3k,2]]\) Campbell-Howard family with a quasi-transversal logical \(CCZ\) gate [19].
\([[8,3,2]]\) Smallest interesting color code

References

[1]
A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
[2]
E. Campbell, “The smallest interesting colour code”, (2016), accessed on 2019-12-09 URL
[3]
Qiskit Community, “Qiskit QEC framework”, URL
[4]
D. Hangleiter, M. Kalinowski, D. Bluvstein, M. Cain, N. Maskara, X. Gao, A. Kubica, M. D. Lukin, and M. J. Gullans, “Fault-Tolerant Compiling of Classically Hard Instantaneous Quantum Polynomial Circuits on Hypercubes”, PRX Quantum 6, (2025) arXiv:2404.19005 DOI
[5]
H. Chen, M. Vasmer, N. P. Breuckmann, and E. Grant, “Automated discovery of logical gates for quantum error correction (with Supplementary (153 pages))”, Quantum Information and Computation 22, 947 (2022) arXiv:1912.10063 DOI
[6]
J. Haah and M. B. Hastings, “Measurement sequences for magic state distillation”, Quantum 5, 383 (2021) arXiv:2007.07929 DOI
[7]
E. J. Kuehnke, K. Levi, J. Roffe, J. Eisert, and D. Miller, “Hardware-tailored logical Clifford circuits for stabilizer codes”, (2025) arXiv:2505.20261
[8]
S. Wu, D. Zhong, T. A. Brun, and D. A. Lidar, “Universal Weakly Fault-Tolerant Quantum Computation via Code Switching in the [[8,3,2]] Code”, (2026) arXiv:2603.15610
[9]
J. S. Nelson, A. J. Landahl, and A. D. Baczewski, “A small and interesting architecture for early fault-tolerant quantum computers”, (2025) arXiv:2507.20387
[10]
Y. Wang et al., “Fault-tolerant one-bit addition with the smallest interesting color code”, Science Advances 10, (2024) arXiv:2309.09893 DOI
[11]
F. Butt, I. Pogorelov, R. Freund, A. Steiner, M. Meyer, T. Monz, and M. Müller, “Demonstration of measurement-free universal fault-tolerant quantum computation”, (2025) arXiv:2506.22600
[12]
D. Honciuc Menendez, A. Ray, and M. Vasmer, “Implementing fault-tolerant non-Clifford gates using the [[8,3,2]] color code”, Physical Review A 109, (2024) arXiv:2309.08663 DOI
[13]
D. Bluvstein et al., “Logical quantum processor based on reconfigurable atom arrays”, Nature 626, 58 (2023) arXiv:2312.03982 DOI
[14]
M. A. Webster, B. J. Brown, and S. D. Bartlett, “The XP Stabiliser Formalism: a Generalisation of the Pauli Stabiliser Formalism with Arbitrary Phases”, Quantum 6, 815 (2022) arXiv:2203.00103 DOI
[15]
M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
[16]
M. A. Webster, A. O. Quintavalle, and S. D. Bartlett, “Transversal diagonal logical operators for stabiliser codes”, New Journal of Physics 25, 103018 (2023) arXiv:2303.15615 DOI
[17]
B. W. Reichardt et al., “Demonstration of quantum computation and error correction with a tesseract code”, (2024) arXiv:2409.04628
[18]
J. M. Koh, A. Gong, A. C. Diaconu, D. B. Tan, A. A. Geim, M. J. Gullans, N. Y. Yao, M. D. Lukin, and S. Majidy, “Entangling logical qubits without physical operations”, (2026) arXiv:2601.20927
[19]
E. T. Campbell and M. Howard, “Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost”, Physical Review A 95, (2017) arXiv:1606.01904 DOI
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Zoo Code ID: stab_8_3_2

Cite as:
“\([[8,3,2]]\) Smallest interesting color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stab_8_3_2
BibTeX:
@incollection{eczoo_stab_8_3_2, title={\([[8,3,2]]\) Smallest interesting color code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/stab_8_3_2} }
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“\([[8,3,2]]\) Smallest interesting color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stab_8_3_2

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/8/stab_8_3_2.yml.