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

Transversal Gates

CZ gates between any two logical qubits [3] and (weakly) transversal \(CCZ\) gate [1–3].

Fault Tolerance

\(CCZ\) gate can be distilled in a fault-tolerant manner [4].

Realizations

Trapped ions: one-qubit addition algorithm implemented fault-tolerantly on the Quantinuum H1-1 device [5].Superconducting circuits: fault-tolerant \(CZZ\) gate performed on IBM and IonQ devices [6].Rydberg atom arrays: Lukin group [7]. 48 logical qubits, 228 logical two-qubit gates, 48 logical \(CCZ\) gates, and error detection peformed in 16 blocks. Circuit outcomes were sampled and cross-entropy (XEB) was calculated to verify quantumness. Logical entanglement entropy was measured [7].

Cousins

\([[15,1,3]]\) quantum Reed-Muller code— The \([[8,3,2]]\) code can be obtained from a subset of physical qubits of the \([[15,1,3]]\) code [8].
3D surface code— The \([[8,3,2]]\) code can be concatenated with a 3D surface code to yield a \([[O(d^3),3,2d]]\) code family that admits a transversal implementation of the logical \(CCZ\) gate [9].
Concatenated qubit code— The \([[8,3,2]]\) code can be concatenated with a 3D surface code to yield a \([[O(d^3),3,2d]]\) code family that admits a transversal implementation of the logical \(CCZ\) gate [9].
\([[10,1,2]]\) CSS code— The \([[10,1,2]]\) CSS code can be obtained by morphing the \([[8,3,2]]\) code [8].
\([[12,2,2]]\) 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 [10].
\([[16,6,4]]\) Tesseract color code— Applying CNOT gates to the tesseract color code disentangles it into two \([[8,3,2]]\) color codes [11].

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

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Qubit Hamiltonian-based QECC Quantum

Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum

Quantum rainbow code

Quantum pin codeStabilizer Hamiltonian-based Qubit QECC Quantum

Color code

3D color codeQubit Generalized homological-product Lattice stabilizer QLDPC CSS 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 codeSelf-complementary qubit Amplitude-damping (AD) AQECC Color Quantum Reed-Muller Generalized homological-product QLDPC CSS Stabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum

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

XS stabilizer codeQubit QECC Quantum

The \([[8,3,2]]\) code can be viewed as an XS stabilizer code [12; Exam. 6.10].

\([[6k+2,3k,2]]\) Campbell-Howard codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum

\([[8,3,2]]\) 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]: 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
[4]: J. Haah and M. B. Hastings, “Measurement sequences for magic state distillation”, Quantum 5, 383 (2021) arXiv:2007.07929 DOI
[5]: Y. Wang et al., “Fault-tolerant one-bit addition with the smallest interesting color code”, Science Advances 10, (2024) arXiv:2309.09893 DOI
[6]: 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
[7]: D. Bluvstein et al., “Logical quantum processor based on reconfigurable atom arrays”, Nature 626, 58 (2023) arXiv:2312.03982 DOI
[8]: M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
[9]: 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 IQP circuits on hypercubes”, (2025) arXiv:2404.19005
[10]: 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
[11]: B. W. Reichardt et al., “Demonstration of quantum computation and error correction with a tesseract code”, (2024) arXiv:2409.04628
[12]: 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

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.