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We demonstrate the implementation of a novel machine learning framework forprobability density estimation and classification using quantum circuits. Theframework maps a training data set or a single data sample to the quantum stateof a physical system through quantum feature maps. The quantum state of thearbitrarily large training data set summarises its probability distribution ina finite-dimensional quantum wave function. By projecting the quantum state ofa new data sample onto the quantum state of the training data set, one canderive statistics to classify or estimate the density of the new data sample.Remarkably, the implementation of our framework on a real quantum device doesnot require any optimisation of quantum circuit parameters. Nonetheless, wediscuss a variational quantum circuit approach that could leverage quantumadvantage for our framework.

Optimisation-free Classification and Density Estimation with Quantum Circuits