Also known as Smallest interesting color code.
Description
Smallest 3D color code whose physical qubits lie on vertices of a cube and which admits a (weakly) transversal CCZ gate.
Transversal Gates
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 [8].
Parents
- 3D color code — The \([[8,3,2]]\) code is the smallest non-trivial 3D color code.
- \([[2^D,D,2]]\) hypercube quantum code — The \([[8,3,2]]\) code is a hypercube code for \(D=3\).
- XS stabilizer code — The \([[8,3,2]]\) code can be viewed as an XS stabilizer code [9; Exam. 6.10].
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 [10].
- 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 [11].
- 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 [11].
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 et al., “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 Colour Code”, (2023) arXiv:2309.09893
- [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]
- Q. Xu et al., “Constant-Overhead Fault-Tolerant Quantum Computation with Reconfigurable Atom Arrays”, (2023) arXiv:2308.08648
- [8]
- D. Bluvstein et al., “Logical quantum processor based on reconfigurable atom arrays”, Nature (2023) arXiv:2312.03982 DOI
- [9]
- 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
- [10]
- M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
- [11]
- D. Hangleiter et al., “Fault-tolerant compiling of classically hard IQP circuits on hypercubes”, (2024) arXiv:2404.19005
Page edit log
- Victor V. Albert (2022-12-03) — most recent
Cite as:
“\([[8,3,2]]\) CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stab_8_3_2