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Existence of Quantum Splines via Fourth-Order Gradient Flows

Chun-Chi Lin, Yang-Kai Lue, Dung The Tran

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.

Existence of Quantum Splines via Fourth-Order Gradient Flows

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.
Paper Structure (6 sections, 3 theorems, 83 equations, 1 figure)

This paper contains 6 sections, 3 theorems, 83 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Quantum Spline Interpolation via Gradient Flows
  3. Solutions to the Gradient Flow
  4. The Preparatory Steps
  5. The Short-Time Existence
  6. The Long-Time Existence and Asymptotics

Key Result

Theorem 2.1

Let $\sigma \in (0,\infty)$ and $\alpha_1, \alpha_2 \in(0,1)$ with $\alpha_1>2 \alpha_2$. Suppose the initial datum $(U_0, V_{1,0}, \ldots, V_{q,0})\in \Theta_{\mathcal{P}}$ satisfies the compatibility conditions of order $0$ for the gradient flow of $(U, V_{1}, \ldots, V_{q})$ in U_flow and V_flow. Furthermore, there exists $(U_{\infty}, V_{1, \infty}, \ldots, V_{q, \infty}) \in \Theta_{\mathcal{

Figures (1)

  • Figure 1: The case of $q=2$.

Theorems & Definitions (8)

  • Definition 2.1: The compatibility conditions of order $0$
  • Theorem 2.1
  • Theorem 3.1: Solutions to the coupled linear parabolic systems, \ref{['eq:higher-order-linear-g_l']}$\sim$\ref{['eq:second-order-linear-h_l']}
  • proof : The Proof of Theorem \ref{['thm:STE_for_linear-fitting']}
  • Theorem 3.2
  • Remark 3.1
  • proof : The Proof of Theorem \ref{['theorem-short-time']}
  • proof : The Proof of Theorem \ref{['thm:Main_Thm_1']}