\([[4,1,2]]_5\) ququint RM code[1]
Description
A \([[4,1,2]]_5\) pure prime-qudit CSS code on four ququints, i.e., five-dimensional qudits, that is the smallest known non-trivial ququint stabilizer code with a transversal non-Clifford gate [1]. Analogous qubit and qutrit quantum RM codes require 15 and 8 qudits, respectively [1].
A stabilizer tableau for the code is [1; Def. 6] \begin{align} \begin{array}{cccc} X & X^{2} & X^{3} & X^{4} \\ Z & Z^{2} & Z^{3} & Z^{4} \\ Z & Z^{4} & Z^{4} & Z \end{array}~. \tag*{(1)}\end{align} The \(X\)-type generator spans the shortened first-order GRM code of length four, and the \(Z\)-type generators span the dual of the code generated by \((1,2,3,4)\) and the all-ones vector [1].
The code admits the codewords \begin{align} |\overline{j}\rangle = \frac{1}{\sqrt{5}} \sum_{a=0}^{4} |a+j,\,2a+j,\,3a+j,\,4a+j\rangle~, \tag*{(2)}\end{align} where \(j\in\mathbb{Z}_5\) labels the logical ququint and where addition is modulo five.
Magic
Magic-state yield parameter \(\gamma = \log_D (n/k) = \log_2 4 = 2\) for distance \(D=2\) [1; Table I].Transversal and Permutation-Based Gates
Logical Pauli operators are \(\overline{X} = X^{\otimes 4}\) and \(\overline{Z} = (Z^{\dagger})^{\otimes 4}\) [1; Def. 6].Applying the diagonal gate \begin{align} M = \mathrm{diag}\left(\omega^{3},\omega,\omega^{-1},\omega^{-2},\omega^{-1}\right)~,\quad\quad \omega = e^{2\pi i/5} \tag*{(3)}\end{align} to every ququint implements the logical \(\overline{M}^{\dagger}\) [1; Sec. IV.A]. This non-Clifford gate lies in the third level of the qudit Clifford hierarchy [1]. It is Clifford-equivalent to the modular-qudit \(T\) gate [2,3].Threshold
Magic-state distillation threshold of \(\epsilon^{*}_{\mathrm{dep}} \approx 0.363\) against depolarizing noise and \(\epsilon^{*} \approx 0.312\) against arbitrary noise, assuming noiseless Clifford operations [1; Sec. V.D]. Both thresholds refer to the error probability \(\epsilon = 1-\langle M_0|\rho|M_0\rangle\) of the input magic states \(|M_0\rangle = M|+\rangle\). The depolarizing-noise threshold corresponds to a depolarizing rate of about \(0.45\) [1; Sec. V.B].Cousin
- \([[15,1,3]]\) quantum RM code— The \([[15,1,3]]\) quantum RM code is the smallest qubit stabilizer code with distance at least three and a strongly transversal non-Clifford gate [4]. Both codes belong to a family of CSS codes on \(p^m-1\) qudits of prime dimension \(p\) whose \(X\)-type stabilizers form shortened first-order GRM codes [1].
Primary Hierarchy
References
- [1]
- 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
- [2]
- M. Howard and J. Vala, “Qudit versions of the qubitπ/8gate”, Physical Review A 86, (2012) arXiv:1206.1598 DOI
- [3]
- E. T. Campbell, “Enhanced Fault-Tolerant Quantum Computing ind-Level Systems”, Physical Review Letters 113, (2014) arXiv:1406.3055 DOI
- [4]
- S. Koutsioumpas, D. Banfield, and A. Kay, “The Smallest Code with Transversal T”, (2022) arXiv:2210.14066
- [5]
- A. Krishna and J.-P. Tillich, “Towards Low Overhead Magic State Distillation”, Physical Review Letters 123, (2019) arXiv:1811.08461 DOI
Page edit log
- Victor V. Albert (2026-09-23) — most recent
- Victor V. Albert (2026-09-22)
Cite as:
“\([[4,1,2]]_5\) ququint RM code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/qudit_4_1_2, arXiv:2606.11484