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

\([[8,3,2]]\) 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 \(CZZ\) 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

\([[15,1,3]]\) quantum RM code— The \([[8,3,2]]\) code can be obtained from a subset of physical qubits of the \([[15,1,3]]\) code [14].
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 [14].
\([[12,2,2]]\) 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 [15].
\([[16,6,4]]\) Tesseract color code— Applying CNOT gates to the tesseract color code disentangles it into two \([[8,3,2]]\) color codes [16].

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 codeQubit CSS 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 CSS Stabilizer Hamiltonian-based Self-complementary qubit Amplitude-damping (AD) Qubit Small-distance block quantum QECC AQECC Quantum

The \([[8,3,2]]\) code is a hypercube code for \(D=3\).

XP stabilizer codeQubit QECC Quantum

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 \(N=8\) [17; Exam. 6.10].

\([[6k+2,3k,2]]\) 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 [18].

\([[8,3,2]]\) 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,” Online available at https://earltcampbell.com/2016/09/26/the-smallest-interesting-colour-code/ (2016), accessed on 2019-12-09
[3]: Qiskit Community. Qiskit QEC framework. https://github.com/qiskit-community/qiskit-qec
[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. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
[15]: 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
[16]: B. W. Reichardt et al., “Demonstration of quantum computation and error correction with a tesseract code”, (2024) arXiv:2409.04628
[17]: 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
[18]: 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

Page edit log

Victor V. Albert (2022-12-03) — most recent

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

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