Synthesis and Arithmetic of Single Qutrit Circuits

Amolak Ratan Kalra; Michele Mosca; Dinesh Valluri

doi:10.22331/q-2025-02-26-1647

Synthesis and Arithmetic of Single Qutrit Circuits

Amolak Ratan Kalra^1,2,3, Michele Mosca^1,4,3, and Dinesh Valluri^1,4

¹Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada
²David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada
³Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada
⁴Dept. of Combinatorics $\&$ Optimization, University of Waterloo, Waterloo, Ontario, Canada

Published:	2025-02-26, volume 9, page 1647
Editor:	Alexander Dalzell
Eprint:	arXiv:2311.08696v4
Doi:	https://doi.org/10.22331/q-2025-02-26-1647
Citation:	Quantum 9, 1647 (2025).

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Abstract

In this paper we study single qutrit circuits consisting of words over the Clifford$+\mathcal{D}$ cyclotomic gate set, where $\mathcal{D}=\text{diag}(\pm\xi^{a},\pm\xi^{b},\pm\xi^{c})$, $\xi$ is a primitive $9$-th root of unity and $a,b,c$ are integers. We characterize classes of qutrit unit vectors $z$ with entries in $\mathbb{Z}[\xi, \frac{1}{\chi}]$ based on the possibility of reducing their smallest denominator exponent (sde) with respect to $\chi := 1 – \xi,$ by acting an appropriate gate in Clifford$+\mathcal{D}$. We do this by studying the notion of `derivatives mod $3$' of an arbitrary element of $\mathbb{Z}[\xi]$ and using it to study the smallest denominator exponent of $H\mathcal{D}z$ where $H$ is the qutrit Hadamard gate and $\mathcal{D}$. In addition, we reduce the problem of finding all unit vectors of a given sde to that of finding integral solutions of a positive definite quadratic form along with some additional constraints. As a consequence we prove that the Clifford$+\mathcal{D}$ gates naturally arise as gates with sde $0$ and $3$ in the group $U(3,\mathbb{Z}[\xi, \frac{1}{\chi}])$ of $3 \times 3$ unitaries with entries in $\mathbb{Z}[\xi, \frac{1}{\chi}]$. We illustrate the general applicability of these methods to obtain an exact synthesis algorithm for Clifford$+R$ and recover the previously known exact synthesis algorithm of Kliuchnikov, Maslov, Mosca (2012). The framework developed to formulate qutrit gate synthesis for Clifford$+\mathcal{D}$ extends to qudits of arbitrary prime power.

► BibTeX data

@article{Kalra2025synthesisarithmetic,
  doi = {10.22331/q-2025-02-26-1647},
  url = {https://doi.org/10.22331/q-2025-02-26-1647},
  title = {Synthesis and {A}rithmetic of {S}ingle {Q}utrit {C}ircuits},
  author = {Kalra, Amolak Ratan and Mosca, Michele and Valluri, Dinesh},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1647},
  month = feb,
  year = {2025}
}

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Cited by

[1] Jacky Jiang, Natalie Klco, and Olivia Di Matteo, "Non-Abelian dynamics on a cube: Improving quantum compilation through qudit-based simulations", Physical Review D 112 7, 074512 (2025).

[2] Ali Therrien-Motamedi and Zeljko Zilic, 2026 IEEE 56th International Symposium on Multiple-Valued Logic (ISMVL) 13 (2026) ISBN:979-8-3315-5956-4.

[3] Erik J. Gustafson, Henry Lamm, Diyi Liu, Edison M. Murairi, and Shuchen Zhu, "Synthesis of single-qutrit circuits from Clifford +R gates", Physical Review A 112 6, 062414 (2025).

[4] Victor V. Albert and Philippe Faist, "Handbook of Error-Correcting Codes", arXiv:2606.11484, (2026).

[5] Doga Murat Kürkçüoglu, Henry Lamm, and Andrea Maestri, "Qudit Gate Decomposition Dependence for Lattice Gauge Theories", arXiv:2410.16414, (2024).

[6] Amolak Ratan Kalra, Manimugdha Saikia, Dinesh Valluri, Sam Winnick, and Jon Yard, "Multi-qutrit exact synthesis", arXiv:2405.08147, (2024).

[7] Mark Deaconu, Nihar Gargava, Amolak Ratan Kalra, Michele Mosca, and Jon Yard, "Buildings for Synthesis with Clifford+R", arXiv:2510.11526, (2025).

[8] Shai Evra and Ori Parzanchevski, "Arithmeticity, thinness and efficiency of qutrit Clifford+T gates", arXiv:2401.16120, (2024).

[9] Andrew N. Glaudell, Neil J. Ross, John van de Wetering, and Lia Yeh, "Exact Synthesis of Multiqutrit Clifford-Cyclotomic Circuits", arXiv:2405.08136, (2024).

[10] Shiroman Prakash and Tanay Saha, "Low Overhead Qutrit Magic State Distillation", Quantum 9, 1768 (2025).

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