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arXiv:1409.4263

Unitary and anti-unitary quantum description of the classical not gate

Gianpiero Cattaneo, G. Conte, Roberto Leporini

Published Sep 12, 2014• 0 citations
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Abstract

We consider the unitary and the anti--unitary operator realizations of two important genuine quantum gates that transform elements of the computational basis of into superpositions: the square root of the identity and the square root of the negation.
DOI
10.48550/arxiv.1409.4263

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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.
The action of a NOT gate on a classical bit results in a change of its value from a 0 to a 1 and vice versa. The action of the classical NOT gate is in principle perfect because with fidelity equal to unity it complements the value of a bit. The action of the quantum NOT gate in a computational basis |0⟩ and |1⟩ is very similar to the action of the classical NOT gate. However, a more general quantum mechanical operation which corresponds to a classical NOT gate would take a qubit in an arbitrary state |Ψ⟩ and produce a qubit in the state |Ψ⊥⟩ orthogonal to |Ψ⟩. This operation is anti-unitary and therefore, cannot be realized exactly. So how well we can do? We find a unitary transformation acting on an input qubit and some auxiliary qubits, which represent degrees of freedom of the quantum NOT gate itself, which approximately realizes the NOT operation on the state of the original qubit. We call this ‘device’ a universal-NOT gate because the size of the error it produces is independent of the input state. We show that an optimal U-NOT gate which has as its input N identical qubits and produces M outputs achieves a fidelity of F = (N + 1)/ (N + 2), which is equal to the fidelity of estimation of the input qubits. We also show that when a priori information about the state of the input qubit is available, the fidelity of a quantum NOT gate can be much better than the fidelity of estimation.
In this paper we present a novel reversible implementation of a square-root circuit with an array structure. In scientific computations such as numerical analysis, computer graphics, complex number computations, square root is an important operation. In classical irreversible arena we find different realizations of square root circuit. Since reversible circuit is emerging as an alternative to classical circuit, here we introduce a novel reversible realization of this operation. As a basic module, we propose a reversible controlled adder/subtractor (RCAS) block based on 2's Complement computation. In our design we use an array of such RCAS blocks which perform addition or subtraction based on the result generated from digit-by-digit square root operation. To our best knowledge this is the first methodical approach for implementing reversible square root circuit. The new structure of the circuit and different parameters - number of gates, garbage bits and quantum cost for n-bit realization is presented here.