Description
Modular-qudit stabilizer code constructed from generalized Reed-Muller (GRM) codes or their duals via the modular-qudit CSS construction. An odd-prime-qudit CSS code family constructed from first-order punctured GRM codes transversally implements a diagonal gate at any level of the qudit Clifford hierarchy [2].
Magic
An odd-prime-qudit CSS code family constructed from first-order punctured GRM codes can be used for qudit magic-state distillation; see [2; Table I] for yields.
Transversal Gates
An odd-prime-qudit CSS code family constructed from first-order punctured GRM codes transversally implements a diagonal gate at any level of the qudit Clifford hierarchy [2].
Parents
- Modular-qudit CSS code
- Galois-qudit quantum RM code — Galois-qudit RM codes reduce to prime-qudit RM codes when \(q\) is prime.
Children
- Quantum Reed-Muller code — Prime-qudit RM codes reduce to quantum RM codes when \(q=p=2\).
- \([[2^r-1, 2^r-2r-1, 3]]_p\) quantum Hamming code — \([[2^r-1, 2^r-2r-1, 3]]_p\) quantum Hamming codes are constructed using \(q\)-ary Hamming codes, which themselves are dual to first-order GRM codes [3; pg. 45].
References
- [1]
- P. K. Sarvepalli and A. Klappenecker, “Nonbinary Quantum Reed-Muller Codes”, (2005) arXiv:quant-ph/0502001
- [2]
- E. T. Campbell, H. Anwar, and D. E. Browne, “Magic-State Distillation in All Prime Dimensions Using Quantum Reed-Muller Codes”, Physical Review X 2, (2012) arXiv:1205.3104 DOI
- [3]
- M. A. Tsfasman and S. G. Vlăduţ, Algebraic-Geometric Codes (Springer Netherlands, 1991) DOI
Page edit log
- Victor V. Albert (2024-03-01) — most recent
Cite as:
“Prime-qudit RM code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/qudit_reed_muller