# Supervised Learning with Quantum Measurements

arXiv: https://arxiv.org/abs/2004.01227 · HTML: https://mlsanity.com/paper/2004.01227/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Apr 2020.

## Built on by (most cited first, 7 of 7)

- [Learning with Density Matrices and Random Features](https://mlsanity.com/paper/2102.04394/) (2021)
- [Non-IID Quantum Federated Learning with One-shot Communication Complexity](https://mlsanity.com/paper/2209.00768/) (2022)
- [InQMAD: Incremental Quantum Measurement Anomaly Detection](https://mlsanity.com/paper/2210.05061/) (2022)
- [Quantum Measurement Classification with Qudits](https://mlsanity.com/paper/2107.09781/) (2021)
- [Optimisation-free Classification and Density Estimation with Quantum Circuits](https://mlsanity.com/paper/2203.14452/) (2022)
- [MEMO-QCD: Quantum Density Estimation through Memetic Optimisation for Quantum Circuit Design](https://mlsanity.com/paper/2406.08591/) (2024)
- [Bayesian statistical learning using density operators](https://mlsanity.com/paper/2212.14715/) (2022)

## Abstract (arXiv)

This paper reports a novel method for supervised machine learning based on the mathematical formalism that supports quantum mechanics. The method uses projective quantum measurement as a way of building a prediction function. Specifically, the relationship between input and output variables is represented as the state of a bipartite quantum system. The state is estimated from training samples through an averaging process that produces a density matrix. Prediction of the label for a new sample is made by performing a projective measurement on the bipartite system with an operator, prepared from the new input sample, and applying a partial trace to obtain the state of the subsystem representing the output. The method can be seen as a generalization of Bayesian inference classification and as a type of kernel-based learning method. One remarkable characteristic of the method is that it does not require learning any parameters through optimization. We illustrate the method with different 2-D classification benchmark problems and different quantum information encodings.