# Quantum Linear Coherent Controller Synthesis: A Linear Fractional Representation Approach

## TL;DR

This paper develops a linear fractional representation method for synthesizing linear coherent quantum controllers, addressing stabilization, control design, and optimization for quantum systems modeled by quantum stochastic differential equations.

## Contribution

It introduces a new frequency domain parameterization of all stabilizing controllers and formulates quantum control problems within this framework, including a gradient descent approach.

## Key findings

- Derived stabilization conditions using stable factorization.
- Parameterization of all stabilizing controllers in the frequency domain.
- Outlined a gradient descent scheme for quantum H_2 control.

## Abstract

This paper is concerned with a linear fractional representation approach to the synthesis of linear coherent quantum controllers for a given linear quantum plant. The plant and controller represent open quantum harmonic oscillators and are modelled by linear quantum stochastic differential equations. The feedback interconnections between the plant and the controller are assumed to be established through quantum bosonic fields. In this framework, conditions for the stabilization of a given linear quantum plant via linear coherent quantum feedback are addressed using a stable factorization approach. The class of all stabilizing quantum controllers is parameterized in the frequency domain. Coherent quantum weighted H_2 and H_\infty control problems for linear quantum systems are formulated in the frequency domain. Finally, a projected gradient descent scheme is outlined for the coherent quantum weighted H_2 control problem.