An Application of Pontryagin Neural Networks to Solve Optimal Quantum Control Problems

TL;DR

This paper introduces a novel approach combining Pontryagin maximum principle with physics-informed neural networks to efficiently solve quantum optimal control problems, specifically minimizing time and energy in quantum state transformations.

Contribution

It formulates a quantum control problem using Pontryagin's principle and applies a new neural network-based numerical method, which is innovative in the quantum control context.

Findings

Neural network approach speeds up quantum control computations

Derived first order optimality conditions for quantum systems

Successfully computes optimal control protocols for quantum states

Abstract

Reliable high-fidelity quantum state transformation has always been considered as an inseparable part of quantum information processing. In this regard, Pontryagin maximum principle has proved to play an important role to achieve the maximum fidelity in an optimum time or energy. Motivated by this, in this work, we formulate a control constrained optimal control problem where we aim to minimize time and also energy subjected to a quantum system satisfying the bilinear Schrodinger equation. We derive the first order optimality conditions through the application of Pontryagin Maximum (minimum) Principle, which results in a boundary value problem. Next, in order to obtain efficient numerical results, we exploit a particular family of physics-informed neural networks that are specifically designed to tackle the indirect method based on the Maximum Principle of Pontryagin. This method has not yet been studied in the quantum context, but it can significantly speed up the process. To this end, we first obtain a set of relations which finally let us compute the optimal control strategy to determine the time- and energy-optimal protocol driving a general initial state to a target state by a quantum Hamiltonian with bounded control. We make use of the so-called "qutip" package in python, and the newly developed "tfc" python package.