Quadratic Form Expansions for Unitaries

TL;DR

This paper presents a novel framework using quadratic form expansions to analyze and efficiently implement unitary operations in quantum computing, linking them to entangled resource states.

Contribution

It introduces techniques to analyze unitaries via quadratic forms and relates these forms to entangled resources, enabling efficient implementation under certain conditions.

Findings

Quadratic form expansions can represent unitaries in quantum computing.

Conditions are identified for efficient implementation of unitaries using quadratic forms.

The relation between quadratic forms and entangled resource states is established.

Abstract

We introduce techniques to analyze unitary operations in terms of quadratic form expansions, a form similar to a sum over paths in the computational basis when the phase contributed by each path is described by a quadratic form over $\mathbb R$. We show how to relate such a form to an entangled resource akin to that of the one-way measurement model of quantum computing. Using this, we describe various conditions under which it is possible to efficiently implement a unitary operation U, either when provided a quadratic form expansion for U as input, or by finding a quadratic form expansion for U from other input data.