# Quantum Gate Generation in Two-Level Open Quantum Systems by Coherent and Incoherent Photons Found with Gradient Search

## TL;DR

This paper develops a gradient-based optimization method to generate quantum gates in open two-level systems by combining coherent and incoherent controls, leveraging spectral density modulation of incoherent photons.

## Contribution

It introduces a novel gradient calculation for control of open quantum systems with both coherent and incoherent controls, enabling efficient optimization of quantum gate generation.

## Key findings

- Optimized control protocols for single-qubit gates in open systems.
- Gradient expressions derived for piecewise constant controls.
- Method applicable to non-unitary processes and multi-level systems in future work.

## Abstract

In this work, we consider an environment formed by incoherent photons as a resource for controlling open quantum systems via an incoherent control. We exploit a coherent control in the Hamiltonian and an incoherent control in the dissipator which induces the time-dependent decoherence rates $\gamma_k(t)$ (via time-dependent spectral density of incoherent photons) for generation of single-qubit gates for a two-level open quantum system which evolves according to the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation with time-dependent coefficients determined by these coherent and incoherent controls. The control problem is formulated as minimization of the objective functional, which is the sum of Hilbert-Schmidt norms between four fixed basis states evolved under the GKSL master equation with controls and the same four states evolved under the ideal gate transformation. The exact expression for the gradient of the objective functional with respect to piecewise constant controls is obtained. Subsequent optimization is performed using a gradient type algorithm with an adaptive step size that leads to oscillating behaviour of the gradient norm vs iterations. Optimal trajectories in the Bloch ball for various initial states are computed. A relation of quantum gate generation with optimization on complex Stiefel manifolds is discussed. We develop methodology and apply it here for unitary gates as a testing example. The next step is to apply the method for generation of non-unitary processes and to multi-level quantum systems.