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IEEE International Symposium on Multiple-Valued Logic (ISMVL)

The Roots of the NOT Gate

Alexis De Vos, Stijn De Baerdemacker

Published May 1, 2012• 5 citations
Publisher

Abstract

The quantum gates called 'k th root of NOT' and 'controlled k th root of NOT' can be applied to synthesize circuits, both classical reversible circuits and quantum circuits. Such circuits, acting on w qubits, fill a (2w-1)2-dimensional subspace of the (2w)2-dimensional space U(2w) of the 2w× 2wunitary matrices and thus describe computers situated between classical reversible computers and full quantum computers.

DOI
10.1109/ismvl.2012.14

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The quantum gates called 'k th root of NOT' and 'controlled k th root of NOT' can be applied to synthesize circuits, both classical reversible circuits and quantum circuits. Such circuits, acting on w qubits, fill a (2w-1)2-dimensional subspace of the (2w)2-dimensional space U(2w) of the 2w× 2wunitary matrices and thus describe computers situated between classical reversible computers and full quantum computers.
As two basic building blocks for any quantum circuit, we consider the 1-qubit PHASOR circuit ϕ( θ ) and the 1-qubit NEGATOR circuit N ( θ ). Both are roots of the IDENTITY circuit. Indeed: both ϕ(0) and N (0) equal the 2 × 2 unit matrix. Additionally, the NEGATOR is a root of the classical NOT gate. Quantum circuits (acting on w qubits) consisting of controlled PHASORs are represented by matrices from ZU(2 w ); quantum circuits consisting of controlled NEGATORs are represented by matrices from XU(2 w ). Here, ZU( n ) and XU( n ) are subgroups of the unitary group U( n ): the group XU( n ) consists of all n × n unitary matrices with all 2 n line sums (i.e. all n row sums and all n column sums) equal to 1 and the group ZU( n ) consists of all n × n unitary diagonal matrices with first entry equal to 1. Any U( n ) matrix can be decomposed into four parts: U = exp( iα ) Z 1 XZ 2 , where both Z 1 and Z 2 are ZU( n ) matrices and X is an XU( n ) matrix. We give an algorithm to find the decomposition. For n = 2 w it leads to a four-block synthesis of an arbitrary quantum computer.
Abstract Quantum gates, equivalent to logic gates, modify the state vectors. Linear algebra calls these gates linear transformations. In quantum language, they are “operators.” These gates include the NOT gate, the Pauli gates, and the Hadamard gate. The operations of the gates are represented both as abstract symbols and by matrices multiplying state vectors (expressed as column vectors). Quantum computers use a series of gates called a quantum circuit. The chapter explains how to read a quantum circuit diagram and translate it into linear algebra mathematics. The mathematical requirements on the matrices allowed to be used is delineated. Gates are reversible by applying the same gate twice, but measurements are not reversible.
One approach to determining a quantum circuit is to first synthesize a circuit composed of binary reversible gates and to then map that circuit to an equivalent quantum gate realization. This paper considers the mapping phase with the goal of reducing the number of quantum gates required. Our method is based on novel line labeling and gate moving procedures. Results are presented for the quantum library: NOT, controlled-NOT, and the square-root-of-NOT gates (V and V+). The approach is applicable to other quantum gate libraries.