# Variational quantum algorithms for nonlinear problems

## TL;DR

This paper introduces a variational quantum algorithm capable of efficiently solving nonlinear problems, such as nonlinear PDEs, by leveraging multiple quantum state copies and tensor networks, with demonstrated advantages over classical methods.

## Contribution

It presents a novel variational quantum algorithm for nonlinear problems, utilizing multiple state copies and tensor networks, with experimental validation on IBM Q.

## Key findings

- Exponential efficiency over matrix product states.
- Successful experimental demonstration on IBM Q.
- Effective handling of nonlinear PDEs with quantum algorithms.

## Abstract

We show that nonlinear problems including nonlinear partial differential equations can be efficiently solved by variational quantum computing. We achieve this by utilizing multiple copies of variational quantum states to treat nonlinearities efficiently and by introducing tensor networks as a programming paradigm. The key concepts of the algorithm are demonstrated for the nonlinear Schr\"{o}dinger equation as a canonical example. We numerically show that the variational quantum ansatz can be exponentially more efficient than matrix product states and present experimental proof-of-principle results obtained on an IBM Q device.