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Interpolating Binary and Multivalued Logical Quantum Gates
Zeno Toffano 1 , François DUBOIS 2

1  Telecom Dept., CentraleSupélec, Gif-sur-Yvette, France,Lab. des Signaux et Systèmes - L2S (UMR8506) - CNRS - U. Paris-Saclay, France
2  LMSSC, CNAM, Paris, France,, Department of Mathematics, University Paris-Sud, Orsay, France

Published: 20 November 2017 by MDPI AG in Proceedings in 4th International Electronic Conference on Entropy and Its Applications
MDPI AG, Volume 2; 10.3390/ecea-4-05006

A method for synthesizing quantum gates is presented based on interpolation methods applied to operators in Hilbert space. Starting from the diagonal forms of specific generating seed operators with non-degenerate eigenvalue spectrum one obtains for arity-one a complete family of logical operators corresponding to all the one-argument logical connectives. Scaling-up to n -arity gates is obtained by using the Kronecker product and unitary transformations. The quantum version of the Fourier transform of Boolean functions is presented and a Reed-Muller decomposition for quantum logical gates is derived. The common control gates can be easily obtained by considering the logical correspondence between the control logic operator and the binary logic operator. A new polynomial and exponential formulation of the Toffoli gate is presented. The method has parallels to quantum gate-T optimization methods using powers of multilinear operator polynomials. The method is then applied naturally to alphabets greater than two for multi-valued logical gates used for quantum Fourier transform, min-max decision circuits and multivalued adders.

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