Found Phys (2009) 39: 559–572 DOI 10.1007/s10701-009-9302-0 The Algebraic Structure of an Approximately Universal System of Quantum Computational Gates Maria Luisa Dalla Chiara · Roberto Giuntini · Hector Freytes · Antonio Ledda · Giuseppe Sergioli Received: 17 February 2009 / Accepted: 17 March 2009 / Published online: 1 April 2009 © Springer Science+Business Media, LLC 2009 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. We study the basic algebraic properties of this system by introducing the notion of Shi-Aharonov quantum computational structure. We show that the quotient of this structure is isomorphic to a structure based on a particular set of complex numbers (the closed disc with center ( 1 2 , 1 2 ) and radius 1 2 ). Keywords Quantum computation · Quantum logic 1 Introduction Classical circuit theory is basically irreversible in the sense that Boolean functions (gates) are generally described as many-to-one: the same output-bits may correspond Dedicated to Pekka Lahti. M.L. Dalla Chiara Dipartimento di Filosoﬁa, Università di Firenze, via Bolognese 52, 50139 Firenze, Italy e-mail: dallachiara@uniﬁ.it R. Giuntini ( ) · H. Freytes · A. Ledda · G. Sergioli Dipartimento di Scienze Pedagogiche e Filosoﬁche, Università di Cagliari, via Is Mirrionis 1, 09123 Cagliari, Italy e-mail: giuntini@unica.it H. Freytes e-mail: hfreytes@gmail.com A. Ledda e-mail: antonio.ledda@inwind.it G. Sergioli e-mail: giuseppe.sergioli@unisoﬁa.it