Introduction to the Pontryagin Maximum Principle for Quantum Optimal Control
While PMP is elegant, applying it to large quantum systems faces hurdles:
The PMP is based on the idea of adjoining a cost functional to the system dynamics, which leads to the definition of a Hamiltonian function. The Hamiltonian function is a combination of the system dynamics and the cost functional, and it encodes the trade-off between the system's performance and the cost of achieving that performance. Introduction to the Pontryagin Maximum Principle for Quantum
This article has provided a comprehensive introduction to the Pontryagin Maximum Principle for quantum optimal control. The PMP is a powerful tool for solving optimal control problems in quantum systems, and its application has shown great promise in optimizing the control of quantum systems.
: For a control to be considered "optimal," it must maximize this Hamiltonian at every single point in time. The PMP is a powerful tool for solving
The gives necessary conditions for optimal control — and it works just as well for Schrödinger’s equation as for a rocket trajectory.
For a dynamical system ( \dotx = f(x, u) ), one introduces a (or adjoint) vector ( p(t) ). The PMP states that for an optimal pair ( (x^ (t), u^ (t)) ), there exists a non-trivial costate satisfying: For a dynamical system ( \dotx = f(x,
In quantum optimal control, the PMP has been applied to optimize the control of quantum systems. The goal of quantum optimal control is to find a control input that steers a quantum system from an initial state to a target state while minimizing a cost functional.
) : PMP introduces a specialized version of the Hamiltonian that combines the system's dynamics with a set of auxiliary variables called .
Introduction to the Pontryagin Maximum Principle for Quantum Optimal Control
While PMP is elegant, applying it to large quantum systems faces hurdles:
The PMP is based on the idea of adjoining a cost functional to the system dynamics, which leads to the definition of a Hamiltonian function. The Hamiltonian function is a combination of the system dynamics and the cost functional, and it encodes the trade-off between the system's performance and the cost of achieving that performance.
This article has provided a comprehensive introduction to the Pontryagin Maximum Principle for quantum optimal control. The PMP is a powerful tool for solving optimal control problems in quantum systems, and its application has shown great promise in optimizing the control of quantum systems.
: For a control to be considered "optimal," it must maximize this Hamiltonian at every single point in time.
The gives necessary conditions for optimal control — and it works just as well for Schrödinger’s equation as for a rocket trajectory.
For a dynamical system ( \dotx = f(x, u) ), one introduces a (or adjoint) vector ( p(t) ). The PMP states that for an optimal pair ( (x^ (t), u^ (t)) ), there exists a non-trivial costate satisfying:
In quantum optimal control, the PMP has been applied to optimize the control of quantum systems. The goal of quantum optimal control is to find a control input that steers a quantum system from an initial state to a target state while minimizing a cost functional.
) : PMP introduces a specialized version of the Hamiltonian that combines the system's dynamics with a set of auxiliary variables called .