# Algorithms for Quantum Control without Discontinuities; Application to the Simultaneous Control of two Qubits

## TL;DR

This paper introduces a control algorithm for finite-dimensional quantum systems that avoids discontinuities, enabling smooth control of quantum states, with specific application to the simultaneous control of two non-interacting spin-1/2 particles in NMR.

## Contribution

It presents a novel method to design smooth control laws for quantum systems with symmetry properties, reducing the control problem to a fully actuated quotient space.

## Key findings

- Control laws are smooth and free of discontinuities.
- Method successfully applied to two-spin NMR control.
- Provides a flexible framework for quantum state manipulation.

## Abstract

We propose a technique to design control algorithms for a class of finite dimensional quantum systems so that the control law does not present discontinuities. The class of models considered admits a group of symmetries which allows us to reduce the problem of control to a quotient space where the control system is `fully actuated'. As a result we can prescribe a desired trajectory which is, to some extent, arbitrary and derive the corresponding control. We discuss the application to the simultaneous control of two non-interacting spin 1/2 particles with different gyromagnetic ratios in zero field NMR in detail. Our method provides a flexible toolbox for the design of control algorithms to drive the state of finite dimensional quantum systems to any desired final configuration with smooth controls.