# Reinforcement learning for semi-autonomous approximate quantum eigensolver

## TL;DR

This paper introduces a reinforcement learning-based protocol to approximate eigenvectors of Hermitian quantum operators, achieving high fidelity with minimal iterations, useful for semi-autonomous quantum devices.

## Contribution

It presents a novel reinforcement learning protocol for approximating eigenvectors of arbitrary Hermitian operators using measurement and feedback in a quantum setting.

## Key findings

- Achieves over 90% fidelity in less than 10 iterations for single-qubit operators.
- Surpasses 98% fidelity in less than 300 iterations for single-qubit operators.
- Obtains eigenvectors with over 89% fidelity in 8000 iterations for two-qubit operators.

## Abstract

The characterization of an operator by its eigenvectors and eigenvalues allows us to know its action over any quantum state. Here, we propose a protocol to obtain an approximation of the eigenvectors of an arbitrary Hermitian quantum operator. This protocol is based on measurement and feedback processes, which characterize a reinforcement learning protocol. Our proposal is composed of two systems, a black box named environment and a quantum state named agent. The role of the environment is to change any quantum state by a unitary matrix $\hat{U}_E=e^{-i\tau\hat{\mathcal{O}}_E}$ where $\hat{\mathcal{O}}_E$ is a Hermitian operator, and $\tau$ is a real parameter. The agent is a quantum state which adapts to some eigenvector of $\hat{\mathcal{O}}_E$ by repeated interactions with the environment, feedback process, and semi-random rotations. With this proposal, we can obtain an approximation of the eigenvectors of a random qubit operator with average fidelity over 90\% in less than 10 iterations, and surpass 98\% in less than 300 iterations. Moreover, for the two-qubit cases, the four eigenvectors are obtained with fidelities above 89\% in 8000 iterations for a random operator, and fidelities of $99\%$ for an operator with the Bell states as eigenvectors. This protocol can be useful to implement semi-autonomous quantum devices which should be capable of extracting information and deciding with minimal resources and without human intervention.