Computational complexity of quantum optimal control landscapes

TL;DR

This paper analyzes the intrinsic complexity of quantum optimal control problems, revealing that certain control tasks are computationally efficient and scale well with system size, suggesting potential for practical large-scale quantum control.

Contribution

It identifies the Hamiltonian-independent component of quantum control complexity and shows it belongs to the class CLOG, indicating low intrinsic complexity.

Findings

Control optimization complexity is in class CLOG.

Scaling of complexity with system dimension is simple.

Quantum control can be efficient with suitable algorithms.

Abstract

We study the Hamiltonian-independent contribution to the complexity of quantum optimal control problems. The optimization of controls that steer quantum systems to desired objectives can itself be considered a classical dynamical system that executes an analog computation. The system-independent component of the equations of motion of this dynamical system can be integrated analytically for various classes of discrete quantum control problems. For the maximization of observable expectation values from an initial pure state and the maximization of the fidelity of quantum gates, the time complexity of the corresponding computation belongs to the class continuous log (CLOG), the lowest analog complexity class, equivalent to the discrete complexity class NC. The simple scaling of the Hamiltonian-independent contribution to these problems with quantum system dimension indicates that with appropriately designed search algorithms, quantum optimal control can be rendered efficient even for large systems.