Quantum-inspired classification based on quantum state discrimination

We present quantum-inspired algorithms for classification tasks inspired by the problem of quantum state discrimination. While these algorithms could be implemented on a quantum computer, we focus here on their classical implementation. The training of some of these classifiers involves semi-definite programming to find the optimal measurement for the quantum state discrimination problem. We examine how the optimal solution for state discrimination behaves for classification tasks. We also present a relaxation of these classifiers that utilizes linear programming (but that can no longer be interpreted as a quantum measurement). Finally, we consider a classifier based on the pretty good measurement (PGM) and show how to implement it using an analog of the so-called Kernel Trick, which allows us to study its performance on any number of copies of the input state. We evaluate these classifiers on the MNIST and MNIST-1D datasets and find that the PGM generally outperforms the other quantum-inspired classifiers, although it does not perform as well as other, classical, classifiers. Our work thus provides a deeper understanding of the relation between quantum state discrimination and quantum machine learning methods, as well as of the use of kernel methods for quantum machine learning.