Quantum optimal control experiments and simulations have successfully manipulated the dynamics of systems ranging from atoms to biomolecules. Surprisingly, these collective works indicate that the effort (i.e., the number of algorithmic iterations) required to find an optimal control field appears to be essentially invariant to the complexity of the system. The present work explores this matter in a series of systematic optimizations of the state-to-state transition probability on model quantum systems with the number of states $N$ ranging from 5 through 100. The optimizations occur over a landscape defined by the transition probability as a function of the control field. Previous theoretical studies on the topology of quantum control landscapes established that they should be free of sub-optimal traps under reasonable physical conditions. The simulations in this work include nearly 5000 individual optimization test cases, all of which confirm this prediction by fully achieving optimal population transfer of at least 99.9% upon careful attention to numerical procedures to ensure that the controls are free of constraints. Collectively, the simulation results additionally show invariance of required search effort to system dimension $N$. This behavior is rationalized in terms of the structural features of the underlying control landscape. The very attractive observed scaling with system complexity may be understood by considering the distance traveled on the control landscape during a search and the magnitude of the control landscape slope. Exceptions to this favorable scaling behavior can arise when the initial control field fluence is too large or when the target final state recedes from the initial state as $N$ increases.
The core problem in optimal control theory applied to quantum systems is to determine the temporal shape of an applied field in order to maximize the expectation of value of some physical observable. The functional which maps the control field into a given value of the observable defines a Quantum Control Landscape (QCL). Studying the topological and structural features of these landscapes is of critical importance for understanding the process of finding the optimal fields required to effectively control the system, specially when external constraints are placed on both the field $ε(t)$ and t…
Exploring pathways towards quantum advantage in quantum chemistry: the case of a molecule with half-Möbius topology
IBM Research Europe – Zurich（IBM 欧洲苏黎世研究中心）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Éidgenössische Technische Hochschule Zürich (ETHZ)（苏黎世联邦理工学院）；University of Regensburg（雷根斯堡大学）；The University of Manchester（曼彻斯特大学）；Oxford University（牛津大学）
Topological invariants play a key role in the characterization of topological states. Due to the existence of exceptional points, it is a great challenge to detect topological invariants in non-Hermitian systems. We put forward a dynamic winding number, the winding of realistic observables in long-time average, for exploring band topology in both Hermitian and non-Hermitian two-band models via a unified approach. We build a concrete relation between dynamic winding numbers and conventional topological invariants. In one-dimension, the dynamical winding number directly gives the conventional wi…
The broad success of theoretical and experimental quantum optimal control is intimately connected to the topology of the underlying control landscape. For several common quantum control goals, including the maximization of an observable expectation value, the landscape has been shown to lack local optima if three assumptions are satisfied: (i) the quantum system is controllable, (ii) the Jacobian of the map from the control field to the evolution operator is full-rank, and (iii) the control field is not constrained. In the case of the observable objective, this favorable analysis shows that th…
Exploring Quantum Control Landscape and Solution Space Complexity through Dimensionality Reduction & Optimization Algorithms
School of Computer Science, University College Dublin（计算机科学学院，都柏林大学学院）；Centre for Quantum Engineering, Science, and Technology, University College Dublin（量子工程、科学与技术中心，都柏林大学学院）；School of Physics, University College Dublin（物理学院，都柏林大学学院）
Exploring nontrivial topology at quantum criticality in a superconducting processor
College of Computer Science and Technology, Zhejiang University（浙江大学计算机科学与技术学院）；School of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, and Zhejiang Key Laboratory of Micro-nano Quantum Chips and Quantum Control, Zhejiang University（浙江大学物理学院、ZJU-杭州全球科创中心及浙江省微纳量子芯片与量子控制重点实验室）；Institute for Advanced Study in Physics and School of Physics, Zhejiang University（浙江大学物理学高等研究院及物理学院）
