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Serdica Journal of Computing

Computing with the Square Root of NOT

Alexis De Vos, Jan De Beule, Leo Storme

Published Jan 11, 2010• 10 citations
Publisher

Abstract

To the two classical reversible 1-bit logic gates, i.e. the identity gate (a.k.a. the follower) and the NOT gate (a.k.a. the inverter), we add an extra gate, the square root of NOT. Similarly, we add to the 24 classical reversible 2-bit circuits, both the square root of NOT and the controlled square root of NOT. This leads to a new kind of calculus, situated between classical reversible computing and quantum computing.

DOI
10.55630/sjc.2009.3.359-370

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To the two classical reversible 1-bit logic gates, i.e. the identity gate (a.k.a. the follower) and the NOT gate (a.k.a. the inverter), we add an extra gate, the square root of NOT. Similarly, we add to the 24 classical reversible 2-bit circuits, both the square root of NOT and the controlled square root of NOT. This leads to a new kind of calculus, situated between classical reversible computing and quantum computing.
Two possible quantum descriptions of the classical Not gate are investigated in the framework of the Hilbert space C2: the unitary and the anti-unitary operator realizations. The two cases are distinguished interpreting the unitary Not as a quantum realization of the classical gate which on a xed orthogonal pair of unit vectors, realizing once for all the classical bits 0 and 1, produces the required transformations 0 => 1 and 1 => 0 (i.e., logical quantum Not). The anti-unitary Not is a quantum realization of a gate which acts as a classical Not on any pair of mutually orthogonal vectors, each of which is a potential realization of the classical bits (i.e., universal quantum Not). Although the latter is not completely positive, one can give an approximated unitary realization of the gate by appending an ancilla. Finally, we consider the unitary and the anti-unitary operator realizations of two important genuine quantum gates that transform elements of the computational basis of C2 into its superpositions: the square root of the identity and the square root of the Not.