Toward a Theory of Phase Transitions in Quantum Control Landscapes

Control landscape phase transitions (CLPTs) occur as abrupt changes in the cost function landscape upon varying a control parameter and can be revealed by nonanalytic points in statistical order parameters. A prime example are quantum speed limits, 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 quantum speed limit 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 piecewise continuous protocols. Our work provides the first steps toward a systematic theory of CLPTs and paves the way for utilizing statistical field theory techniques for generic complex control landscapes.