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Open Systems & Information Dynamics

The NEGATOR as a Basic Building Block for Quantum Circuits

Alexis De Vos, Stijn De Baerdemacker

Published Mar 1, 2013• 17 citations
Publisher

Abstract

Between (classical) reversible computation and quantum computation there exists an intermediate computational world, represented by unitary matrices that have all line sums equal to 1. All of these quantum circuits can be synthesized with the help of merely two building blocks: the NEGATOR and the singly controlled square root of NOT.
DOI
10.1142/s1230161213500042

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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.