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DOI: 10.4230/LIPIcs.TQC.2022.12
Qutrit Metaplectic Gates Are a Subset of Clifford+T

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

Published in: LIPIcs, Volume 232, 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)

Abstract
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 ⊗ 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.
Cite as

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 für Informatik (2022)

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@InProceedings{glaudell_et_al:LIPIcs.TQC.2022.12,
  author =	{Glaudell, Andrew N. and Ross, Neil J. and van de Wetering, John and Yeh, Lia},
  title =	{{Qutrit Metaplectic Gates Are a Subset of Clifford+T}},
  booktitle =	{17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)},
  pages =	{12:1--12:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-237-2},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{232},
  editor =	{Le Gall, Fran\c{c}ois and Morimae, Tomoyuki},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2022.12},
  URN =		{urn:nbn:de:0030-drops-165195},
  doi =		{10.4230/LIPIcs.TQC.2022.12},
  annote =	{Keywords: Quantum computation, qutrits, gate synthesis, metaplectic gate, Clifford+T}
}
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