Variational Quantum Algorithms for Euclidean Discrepancy and Covariate-Balancing

TL;DR

This paper explores the application of variational quantum algorithms to discrepancy minimization problems, demonstrating that VQE and QAOA can achieve results comparable to classical algorithms in covariate balancing tasks.

Contribution

It introduces a novel quantum approach to discrepancy problems by framing them as quantum Ising models and applying VQA methods.

Findings

VQE and QAOA produce results comparable to classical GSW algorithms.

Quantum algorithms can effectively address Euclidean discrepancy and covariate balancing.

Simulation on IBM quantum hardware shows promising potential for quantum discrepancy algorithms.

Abstract

Algorithmic discrepancy theory seeks efficient algorithms to find those two-colorings of a set that minimize a given measure of coloring imbalance in the set, its {\it discrepancy}. The {\it Euclidean discrepancy} problem and the problem of balancing covariates in randomized trials have efficient randomized algorithms based on the Gram-Schmidt walk (GSW). We frame these problems as quantum Ising models, for which variational quantum algorithms (VQA) are particularly useful. Simulating an example of covariate-balancing on an IBM quantum simulator, we find that the variational quantum eigensolver (VQE) and the quantum approximate optimization algorithm (QAOA) yield results comparable to the GSW algorithm.