Generalized Boolean Functions and Quantum Circuits on IBM-Q

TL;DR

This paper establishes a mathematical link between IBM quantum circuits and multivariate quadratic polynomials over integers modulo 8, enabling a new algebraic perspective on quantum gate operations.

Contribution

It introduces a novel connection between quantum circuits and quadratic polynomials, specifically deriving polynomial representations for Swap and Toffoli gates on IBM-Q.

Findings

Quantum circuits can be expressed as generalized Walsh-Hadamard transforms.

Polynomial representations for Swap and Toffoli gates are explicitly derived.

Provides an algebraic framework for analyzing IBM-Q quantum circuits.

Abstract

We explicitly derive a connection between quantum circuits utilising IBM's quantum gate set and multivariate quadratic polynomials over integers modulo 8. We demonstrate that the action of a quantum circuit over input qubits can be written as generalized Walsh-Hadamard transform. Here, we derive the polynomials corresponding to implementations of the Swap gate and Toffoli gate using IBM-Q gate set.