# A Quantum Kalman Filter-Based PID Controller

## TL;DR

This paper develops a quantum version of a PID controller using a quantum Kalman filter, providing a framework for applying classical control methods to quantum systems under continuous measurement.

## Contribution

It introduces a concrete quantum stochastic model with a PID controller based on filtered estimates, extending classical control techniques to quantum systems.

## Key findings

- Constructs a controlled quantum stochastic dynamical model.
- Uses input and output pictures to relate internal dynamics and measurements.
- Provides a foundation for quantum control theory with classical control methods.

## Abstract

We give a concrete description of a controlled quantum stochastic dynamical model corresponding to a quantum system (a cavity mode) under going continual quadrature measurements, with a PID controller acting on the filtered estimate for the mode operator. Central use is made of the input and output pictures when constructing the model: these unitarily equivalent pictures are presented in the paper, and used to transfer concepts relating to the controlled internal dynamics to those relating to measurement output, and vice versa. The approach shows the general principle for investigating mathematically and physically consistent models in which standard control theoretic methods are to be extended to the quantum setting.