Quantum gradient descent and Newton's method for constrained polynomial optimization

TL;DR

This paper develops quantum versions of gradient descent and Newton's method for constrained polynomial optimization, leveraging quantum algorithms to potentially improve efficiency in high-dimensional problems with few iterations.

Contribution

It introduces quantum algorithms for iterative optimization methods applied to polynomial problems with constraints, utilizing quantum phase estimation and matrix operations.

Findings

Quantum algorithms perform polylogarithmically in problem dimension.

Exponential speedup in the number of iterations for certain problems.

Suitable for high-dimensional problems with limited iterations.

Abstract

Optimization problems in disciplines such as machine learning are commonly solved with iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton's method takes into account curvature information and thereby often improves convergence. Here, we develop quantum versions of these iterative optimization algorithms and apply them to polynomial optimization with a unit norm constraint. In each step, multiple copies of the current candidate are used to improve the candidate using quantum phase estimation, an adapted quantum principal component analysis scheme, as well as quantum matrix multiplications and inversions. The required operations perform polylogarithmically in the dimension of the solution vector and exponentially in the number of iterations. Therefore, the quantum algorithm can be beneficial for high-dimensional problems where a small number of iterations is sufficient.