Toward a Theory of Phase Transitions in Quantum Control Landscapes

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.

The data associated with this manuscript version is available under DOI: 10.5281/zenodo.16900733

References in corpus (15)

• The Magnus expansion and some of its applications
• Modeling and Control of Quantum Systems: An Introduction
• A Landscape Analysis of Constraint Satisfaction Problems
• Dynamical Quantum Error Correction of Unitary Operations with Bounded Controls
• Introduction to Theoretical and Experimental aspects of Quantum Optimal Control
• Disordered Systems Insights on Computational Hardness
• Predicting quantum dynamical cost landscapes with deep learning
• Recursive greedy initialization of the quantum approximate optimization algorithm with guaranteed improvement
• A continuous constraint satisfaction problem for the rigidity transition in confluent tissues
• Introduction to the dynamics of disordered systems: equilibrium and gradient descent
• Control of an environmental spin defect beyond the coherence limit of a central spin
• Optimal control of a quantum sensor: A fast algorithm based on an analytic solution
• Phased Geometric Controls of V-Shaped Three-Level System for Zero-field Quantum Sensing
• Quantum control without quantum states
• Topological Phase Transitions in a Constrained Two-Qubit Quantum Control Landscape

Cited by in corpus (7)

• Optimal Phase-Insensitive Force Sensing with Non-Gaussian States
• Topological Phase Transitions in a Constrained Two-Qubit Quantum Control Landscape
• Fighting Exponentially Small Gaps by Counterdiabatic Driving
• Engineering the uncontrollable: Steering noisy spin-correlated radical-pairs with coherent and incoherent control
• Quantum Dynamical Signatures of Topological Flow Transitions in Limit Cycle Phases
• Optimal Work Extraction from Finite-Time Closed Quantum Dynamics
• Diverse efficiency of observable optimization for four-level quantum systems with higher-order traps