Description
Modular-qudit stabilizer code constructed from 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 [2,3].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 [2,3].Primary Hierarchy
Parents
Galois-qudit RM codes reduce to prime-qudit RM codes when \(q\) is prime.
Prime-qudit RM code
Children
Prime-qudit RM codes reduce to quantum RM codes when \(q=p=2\).
The \([[2^r-1, 2^r-2r-1, 3]]_p\) quantum Hamming code family extends the qubit quantum Hamming family to prime qudits using local-dimension-invariant representations [4].
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]
- E. T. Campbell, “Enhanced Fault-Tolerant Quantum Computing ind-Level Systems”, Physical Review Letters 113, (2014) arXiv:1406.3055 DOI
- [4]
- A. J. Moorthy and L. G. Gunderman, “Local-dimension-invariant Calderbank-Shor-Steane Codes with an Improved Distance Promise”, (2021) arXiv:2110.11510
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