Variational Quantum Algorithms for Euclidean Discrepancy and Covariate-Balancing
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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 discrepancy. The 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.
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