Existence of Quantum Splines via Fourth-Order Gradient Flows

TL;DR

This paper provides a rigorous mathematical foundation for quantum splines, demonstrating their existence via a fourth-order gradient flow approach within a geometric framework, thus enabling controlled quantum evolution modeling.

Contribution

It establishes the first rigorous existence theory for quantum splines using a novel fourth-order gradient flow method tailored to quantum control problems.

Findings

Existence of quantum splines is proven mathematically.

A constructive procedure for generating quantum splines is developed.

The framework adapts to boundary conditions specific to quantum control.

Abstract

We establish a rigorous existence theory for the quantum splines introduced by Brody, Holm, and Meier in Physical Review Letters (2012). These curves arise as solutions of a variational problem on the unitary group describing optimally controlled quantum evolutions. By formulating the problem within a geometric gradient flow framework for Riemannian spline interpolation, we construct a well-posed fourth order evolution whose asymptotic limits realize the desired quantum splines. The analysis requires adapting the variational structure to boundary conditions dictated by the physical model, which are not directly amenable to the setting in our recently developed framework for gradient flows of Riemannian spline interpolation. We show that, despite these difficulties, the modified system admits a rigorous analytical treatment, yielding both existence and a constructive procedure for generating quantum splines. Our results provide a mathematical foundation for the variational description of smooth quantum control trajectories and clarify the analytical structure underlying their formation.