Optimizing a polynomial function on a quantum processor

Keren Li,Guilu Long,Shijie Wei,Feihao Zhang,Patrick Rebentrost,Tao Xin,Pan Gao,Zengrong Zhou,Xiaoting Wang

doi:10.1038/s41534-020-00351-5

Abstract

The gradient descent method is central to numerical optimization and is the key ingredient in many machine learning algorithms. It promises to find a local minimum of a function by iteratively moving along the direction of the steepest descent. Since for high-dimensional problems the required computational resources can be prohibitive, it is desirable to investigate quantum versions of the gradient descent, such as the recently proposed (Rebentrost et al.1). Here, we develop this protocol and implement it on a quantum processor with limited resources. A prototypical experiment is shown with a four-qubit nuclear magnetic resonance quantum processor, which demonstrates the iterative optimization process. Experimentally, the final point converged to the local minimum with a fidelity >94%, quantified via full-state tomography. Moreover, our method can be employed to a multidimensional scaling problem, showing the potential to outperform its classical counterparts. Considering the ongoing efforts in quantum information and data science, our work may provide a faster approach to solving high-dimensional optimization problems and a subroutine for future practical quantum computers.

Highlights

A basic situation in optimization is the minimization or maximization of a polynomial subject to some constraints
As a log ðKpÞ-qubit-controlled gate can be implemented with Oðlog ðKpÞ2 ́ log ðNÞÞ basic quantum gates, which is included in Supplementary Note I(C)[37], OðKplog ðKpÞ2 ́ log ðNÞrÞ basic quantum gates are required for both circuits for r iterations
The protocol provides a quantum circuit only comprised of standard quantum gates, it can in principle be realized in current technologies

Summary

Introduction

A basic situation in optimization is the minimization or maximization of a polynomial subject to some constraints. Optimization, i.e., maximization or minimization, of a cost function can be attempted by the prototypical gradient algorithm iteratively. We develop the gradient algorithm and propose an experimental protocol to perform the gradient descent iterations, with a prototypical experiment to demonstrate the process to optimize polynomials on a quantum simulator.

Results

Conclusion