The Toffoli-Hadamard gate system: an algebraic approach

Dalla Chiara M. L;LEDDA, ANTONIO;SERGIOLI, GIUSEPPE;GIUNTINI, ROBERTO

2013-01-01

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Abstract

Shi and Aharonov have shown that the Toffoli gate and the Hadamard gate give rise to an approximately universal set of quantum computational gates. The basic algebraic properties of this system have been studied in Dalla Chiara etal. (Foundations of Physics 39(6):559–572,2009), where we have introduced the notion of Shi-Aharonov quantum computational structure. In this paper we proposean algebraic abstraction from the Hilbert-space quantum computational structures, by introducing the notion of Toffoli-Hadamard algebra. From an intuitive point of view, such abstract algebras represent a natural quantum generalization of both classical and fuzzy-like structures.