\([[15, 7, 3]]\) Hamming-based CSS code

\([[15, 7, 3]]\) Hamming-based CSS code[1,2]

Description

Self-dual Hamming-based CSS code that admits permutation-based CZ logical gates.

Transversal Gates

Single-qubit Clifford operations applied transversally yield the corresponding Clifford gates on one of the logical qubits [3]. CNOT gate because it is a CSS code. Transversal CCZ gate [4].

Gates

CZ gates can be performed using qubit permutations, and a CCZ gate can be performed using four ancilla qubits [3].

Fault Tolerance

Clifford gates can be performed fault-tolerantly using two ancillary flag qubits, and a CCZ gate can be performed using four ancilla qubits [3].

Parent

\([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS code

Cousin

Perfect quantum code — \([[15, 7, 3]]\) Hamming-based CSS code is perfect as a CSS code, i.e., the number of its \(Z\)-type syndromes matches the number of \(X\)-type Pauli errors up to weight one [3].

References

[1]: A. M. Steane, “Simple quantum error-correcting codes”, Physical Review A 54, 4741 (1996) arXiv:quant-ph/9605021 DOI
[2]: Jim Harrington and Ben W. Reichardt, “Addressable multi-qubit logic via permutations,” Talk at Southwest Quantum Information and Technology (SQuInT) (2011).
[3]: R. Chao and B. W. Reichardt, “Fault-tolerant quantum computation with few qubits”, npj Quantum Information 4, (2018) arXiv:1705.05365 DOI
[4]: A. Paetznick and B. W. Reichardt, “Universal Fault-Tolerant Quantum Computation with Only Transversal Gates and Error Correction”, Physical Review Letters 111, (2013) arXiv:1304.3709 DOI

Page edit log

Victor V. Albert (2022-12-06) — most recent

Cite as:

“\([[15, 7, 3]]\) Hamming-based CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stab_15_7_3

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/small_distance/small/stab_15_7_3.yml.