Qutrit Metaplectic Gates Are a Subset of Clifford+T

Authors Andrew N. Glaudell , Neil J. Ross , John van de Wetering , Lia Yeh

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Author Details

Andrew N. Glaudell
  • Booz Allen Hamilton, Atlanta, GA, USA
  • Department of Mathematics, George Mason University, Fairfax, VA, USA
Neil J. Ross
  • Department of Mathematics and Statistics, Dalhousie University, Halifax, Canada
John van de Wetering
  • Radboud University Nijmegen, The Netherlands
  • University of Oxford, UK
Lia Yeh
  • Department of Computer Science, University of Oxford, UK


We would like to thank Alex Christopher Lim and Anikait Mundhra for creating a Python implementation of the constructions in this paper, available at https://github.com/lia-approves/qudit-circuits/tree/main/qutrit_R_from_T.

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Andrew N. Glaudell, Neil J. Ross, John van de Wetering, and Lia Yeh. Qutrit Metaplectic Gates Are a Subset of Clifford+T. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum fΓΌr Informatik (2022)


A popular universal gate set for quantum computing with qubits is Clifford+T, as this can be readily implemented on many fault-tolerant architectures. For qutrits, there is an equivalent T gate, that, like its qubit analogue, makes Clifford+T approximately universal, is injectable by a magic state, and supports magic state distillation. However, it was claimed that a better gate set for qutrits might be Clifford+R, where R = diag(1,1,-1) is the metaplectic gate, as certain protocols and gates could more easily be implemented using the R gate than the T gate. In this paper we show that the qutrit Clifford+R unitaries form a strict subset of the Clifford+T unitaries when we have at least two qutrits. We do this by finding a direct decomposition of R βŠ— 𝕀 as a Clifford+T circuit and proving that the T gate cannot be exactly synthesized in Clifford+R. This shows that in fact the T gate is more expressive than the R gate. Moreover, we additionally show that it is impossible to find a single-qutrit Clifford+T decomposition of the R gate, making our result tight.

Subject Classification

ACM Subject Classification
  • Theory of computation β†’ Quantum computation theory
  • Quantum computation
  • qutrits
  • gate synthesis
  • metaplectic gate
  • Clifford+T


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