CSS-T code[1] 

Description

A CSS code for which a physical transversal \(T\) gate is either the identity (up to a global phase) or a logical gate. CSS-T codes are constructed from a pair of linear binary codes via the CSS construction, with the pair satisfying certain conditions [2].

Rate

Asymptotically good CSS-T codes exist [3].

Transversal Gates

A physical transversal \(T\) gate is either the identity (up to a global phase) or a logical gate [4].

Parent

Cousins

  • Linear binary code — CSS-T codes are constructed from a pair of linear binary codes via the CSS construction, with the pair satisfying certain conditions [2].
  • Cyclic linear binary code — Binary cyclic and extended cyclic codes can be used to construct CSS-T codes via the CSS construction [2].
  • Quantum Reed-Muller code — Certain quantum RM codes are CSS-T codes [2,5,6].
  • Triorthogonal code — Triorthogonal and CSS-T codes overlap, but neither is a subset of the other [7]. CSS-T codes reduce to triorthogonal codes when the logical action of the physical transversal \(T\) gate is a logical \(T\) gate on all encoded qubits. Triorthogonality is necessary for physical transversal \(T\) gates on each qubit to realize the identity logical gate [4; Thm. 12]. The \(X\)-type stabilizer generator matrix for a CSS-T code is always triorthogonal [2; Corr. 5].

References

[1]
N. Rengaswamy, R. Calderbank, M. Newman, and H. D. Pfister, “Classical Coding Problem from Transversal T Gates”, 2020 IEEE International Symposium on Information Theory (ISIT) (2020) arXiv:2001.04887 DOI
[2]
E. Camps-Moreno, H. H. López, G. L. Matthews, D. Ruano, R. San-José, and I. Soprunov, “An algebraic characterization of binary CSS-T codes and cyclic CSS-T codes for quantum fault tolerance”, Quantum Information Processing 23, (2024) arXiv:2312.17518 DOI
[3]
E. Berardini, R. Dastbasteh, J. E. Martinez, S. Jain, and O. S. Larrarte, “Asymptotically good CSS-T codes exist”, (2024) arXiv:2412.08586
[4]
N. Rengaswamy, R. Calderbank, M. Newman, and H. D. Pfister, “On Optimality of CSS Codes for Transversal T”, IEEE Journal on Selected Areas in Information Theory 1, 499 (2020) arXiv:1910.09333 DOI
[5]
E. Andrade, J. Bolkema, T. Dexter, H. Eggers, V. Luongo, F. Manganiello, and L. Szramowski, “CSS-T Codes from Reed Muller Codes for Quantum Fault Tolerance”, (2023) arXiv:2305.06423
[6]
E. Berardini, A. Caminata, and A. Ravagnani, “Structure of CSS and CSS-T quantum codes”, Designs, Codes and Cryptography (2024) arXiv:2310.16504 DOI
[7]
S. P. Jain and V. V. Albert, “High-distance codes with transversal Clifford and T-gates”, (2024) arXiv:2408.12752
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Zoo Code ID: css-t

Cite as:
“CSS-T code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/css-t
BibTeX:
@incollection{eczoo_css-t, title={CSS-T code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/css-t} }
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Permanent link:
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Cite as:

“CSS-T code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/css-t

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/magic/k-orthogonal/css-t.yml.