# Interpolation Methods for Binary and Multivalued Logical Quantum Gate Synthesis

## TL;DR

This paper introduces interpolation-based methods for synthesizing binary and multivalued quantum logical gates, including new formulations for complex gates like Toffoli, with applications to quantum Fourier transforms and multi-valued logic.

## Contribution

It presents a novel interpolation approach for quantum gate synthesis, extending to multi-valued logic and deriving new formulations for complex gates like Toffoli.

## Key findings

- Complete family of one-argument logical operators derived
- Quantum Fourier transform for multi-valued logic developed
- New polynomial and exponential formulations for Toffoli gate

## 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 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 propositional 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.