https://doi.org/10.1140/epjp/s13360-025-06394-x
Regular Article
Clifford equivalence of Howard–Vala T-gates
1
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190, Beijing, China
2
School of Mathematical Sciences, University of Chinese Academy of Sciences, 100049, Beijing, China
Received:
13
December
2024
Accepted:
1
May
2025
Published online:
5
June
2025
The qubit T-gate plays a pivotal role in the stabilizer formalism of quantum computation, with the paradigm “Clifford + T” serving as a standard framework for universal quantum circuits. In higher dimensional systems, the important Howard–Vala T-gates extend the concept of the qubit T-gate to qudit cases and offer certain optimality in the generation of magic resource from stabilizer states. Since the Howard–Vala T-gates constitute a three-parameter family of quantum gates, it is desirable to study the relations between these various T-gates and identify some representative ones among them. In this work, we investigate the Clifford equivalence of the Howard–Vala T-gates in any prime dimensional quantum system. We completely characterize the structure of Clifford equivalence for these T-gates and reveal its subtle dependence on the arithmetic properties of the system dimension d. We show that when 
or 
(mod 3), all the Howard–Vala T-gates are Clifford equivalent. In contrast, when 
(mod 3), these T-gates can be partitioned into three equivalence classes. These results highlight an intriguing feature of the Howard–Vala T-gates and an intrinsic connection between Clifford equivalence and cubic residues in modular arithmetic.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
