Search complexity and resource scaling for the quantum optimal control of unitary transformations

TL;DR

This paper investigates how the complexity and resource requirements for quantum control of unitary transformations scale with system size, providing methods to classify quantum systems based on control effort.

Contribution

It introduces Hamiltonian-dependent methods to quantify and classify the control complexity of quantum systems for implementing arbitrary unitaries.

Findings

Local extrema in control landscapes have null measure, aiding local search convergence.

Control time scales exponentially with Hilbert space dimension in some systems.

Control resource requirements vary depending on the system Hamiltonian.

Abstract

The optimal control of unitary transformations is a fundamental problem in quantum control theory and quantum information processing. The feasibility of performing such optimizations is determined by the computational and control resources required, particularly for systems with large Hilbert spaces. Prior work on unitary transformation control indicates that (i) for controllable systems, local extrema in the search landscape for optimal control of quantum gates have null measure, facilitating the convergence of local search algorithms; but (ii) the required time for convergence to optimal controls can scale exponentially with Hilbert space dimension. Depending on the control system Hamiltonian, the landscape structure and scaling may vary. This work introduces methods for quantifying Hamiltonian-dependent and kinematic effects on control optimization dynamics in order to classify quantum systems according to the search effort and control resources required to implement arbitrary unitary transformations.