On nonlinear transformations in quantum computation

TL;DR

This paper develops quantum subroutines for implementing nonlinear transformations of quantum states, which could enhance applications like solving nonlinear equations and quantum machine learning.

Contribution

It introduces a framework for nonlinear transformations in quantum computing using weighted states and combines quantum circuits with classical post-processing.

Findings

Provides algorithms for nonlinear state transformations

Framework based on weighted states and classical post-processing

Potential applications in quantum machine learning and nonlinear problem solving

Abstract

While quantum computers are naturally well-suited to implementing linear operations, it is less clear how to implement nonlinear operations on quantum computers. However, nonlinear subroutines may prove key to a range of applications of quantum computing from solving nonlinear equations to data processing and quantum machine learning. Here we develop a series of basic subroutines for implementing nonlinear transformations of input quantum states. Our algorithms are framed around the concept of a weighted state, a mathematical entity describing the output of an operational procedure involving both quantum circuits and classical post-processing.