Interpolation Methods for Binary and Multivalued Logical Quantum Gate Synthesis

Abstract : 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 family of logical operators corresponding to 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 function is presented and a method for Reed-Muller decomposition 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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Zeno Toffano, François Dubois. Interpolation Methods for Binary and Multivalued Logical Quantum Gate Synthesis. TQC2017 - Theory of Quantum Computation, Communication and Cryptography, UPMC, Jun 2017, Paris, France. ⟨hal-01490947v2⟩

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