Toward a Theory of Phase Transitions in Quantum Control Landscapes

TL;DR

This paper develops an analytical framework to understand control landscape phase transitions in quantum systems, linking abrupt changes in control cost functions to topological and geometric properties of optimal protocols.

Contribution

It introduces a systematic theory of control landscape phase transitions using Dyson, Magnus, and cumulant expansions, and relates these to structural changes in optimal control protocols.

Findings

CLPTs are associated with different types of instabilities in optimal protocols.

The topological properties of the set of optimal protocols influence the critical scaling at QSL.

Numerical algorithms confirm the theoretical predictions in single- and two-qubit control problems.

Abstract

Control landscape phase transitions (CLPTs) occur as abrupt changes in the cost function landscape upon varying a control parameter, and can be revealed by non-analytic points in statistical order parameters. A prime example are quantum speed limits (QSL) which mark the onset of controllability as the protocol duration is increased. Here we lay the foundations of an analytical theory for CLPTs by developing Dyson, Magnus, and cumulant expansions for the cost function that capture the behavior of CLPTs with a controlled precision. Using linear and quadratic stability analysis, we reveal that CLPTs can be associated with different types of instabilities of the optimal protocol. This allows us to explicitly relate CLPTs to critical structural rearrangements in the extrema of the control landscape: utilizing path integral methods from statistical field theory, we trace back the critical scaling of the order parameter at the QSL to the topological and geometric properties of the set of optimal protocols, such as the number of connected components and its dimensionality. We verify our predictions by introducing a numerical sampling algorithm designed to explore this optimal set via a homotopic stochastic update rule. We apply this new toolbox explicitly to analyze CLPTs in the single- and two-qubit control problems whose landscapes are analytically tractable, and compare the landscapes for bang-bang and continuous protocols. Our work provides the first steps towards a systematic theory of CLPTs and paves the way for utilizing statistical field theory techniques for generic complex control landscapes.