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2020

Efficient quantum gate teleportation in higher dimensions

de Silva, Nadish

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The Clifford hierarchy is a nested sequence of sets of quantum gates critical to achieving fault-tolerant quantum computation.

Diagonal gates of the Clifford hierarchy and 'nearly diagonal' semi-Clifford gates are particularly important: they admit efficient gate teleportation protocols that implement these gates with fewer ancillary quantum resources such as magic states.
Despite the practical importance of these sets of gates, many questions about their structure remain open; this is especially true in the higher-dimensional qudit setting.
Our contribution is to leverage the discrete Stone-von Neumann theorem and the symplectic formalism of qudit stabiliser mechanics towards extending results of Zeng-Cheng-Chuang (2008) and Beigi-Shor (2010) to higher dimensions in a uniform manner.

Efficient quantum gate teleportation in higher dimensions · Around

Built on

Linear transformations in Hilbert space: III. Operational methods and group theory
M. H. Stone · 1930
Earlier work this paper cites.
Die eindeutigkeit der Schrödingerschen operatoren
J. von Neumann · 1931
Earlier work this paper cites.
A theorem of Stone and von Neumann
G. W. Mackey · 1949
Earlier work this paper cites.
Large abelian subgroups of Chevalley groups
M. J. J. Barry · 1979
Earlier work this paper cites.
Stabilizer Codes and Quantum Error Correction
D. Gottesman · 1997
Earlier work this paper cites.

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Then

Semi-Clifford operations, structure of
B. Zeng, X. Chen, and I. L. Chuang · 2008
Later among the works it cites.
𝒞 3 \mathcal{C}_{3} , semi-Clifford and generalized semi-Clifford operations
S. Beigi and P. W. Shor · 2010
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Magic-state distillation in all prime dimensions using quantum Reed-Muller codes
E. T. Campbell, H. Anwar, and D. E. Browne · 2012
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Quantum Theory for Mathematicians
B. C. Hall · 2013
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Diagonal gates in the Clifford hierarchy
S. X. Cui, D. Gottesman, and A. Krishna · 2017
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