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Wu R.-B.,Tsinghua University | Wu R.-B.,Center for Quantum Information Science and Technology | Long R.,Princeton University | Dominy J.,Princeton University | And 2 more authors.
Physical Review A - Atomic, Molecular, and Optical Physics | Year: 2012

Quantum control landscape theory was formulated to assess the ease of finding optimal control fields in simulations and in the laboratory. The landscape is the observable as a function of the controls, and a primary goal of the theory is the analysis of landscape features. In what is referred to as the kinematic picture of the landscape, prior work showed that the landscapes are generally free of traps that could halt the search for an optimal control at a suboptimal observable value. The present paper considers the dynamical picture of the landscape, seeking the existence of singular controls, especially of a nonkinematic nature along with an assessment of whether they correspond to traps. We analyze the necessary and sufficient conditions for singular controls to be kinematic or nonkinematic critical solutions and the likelihood of their being encountered while maximizing an observable. An algorithm is introduced to seek singular controls on the landscape in simulations along with an associated Hessian landscape analysis. Simulations are performed for a large number of model finite-level quantum systems, showing that all the numerically identified kinematic and nonkinematic singular critical controls are not traps, in support of the prior empirical observations on the ease of finding high-quality optimal control fields. © 2012 American Physical Society.

Li D.,Tsinghua University | Li D.,Center for Quantum Information Science and Technology
Laser Physics Letters | Year: 2016

The adiabatic theorem was proposed about 90 years ago and has played an important role in quantum physics. The quantitative adiabatic condition constructed from eigenstates and eigenvalues of a Hamiltonian is a traditional tool to estimate adiabaticity and has proven to be the necessary and sufficient condition for adiabaticity. However, recently the condition has become a controversial subject. In this paper, we list some expressions to estimate the validity of the adiabatic approximation. We show that the quantitative adiabatic condition is invalid for the adiabatic approximation via the Euclidean distance between the adiabatic state and the evolution state. Furthermore, we deduce general necessary and sufficient conditions for the validity of the adiabatic approximation by different definitions. © 2016 Astro Ltd.

Wu R.-B.,Tsinghua University | Wu R.-B.,Center for Quantum Information Science and Technology | Hsieh M.A.,University of Southern California | Rabitz H.,Princeton University
Physical Review A - Atomic, Molecular, and Optical Physics | Year: 2011

This paper reveals an important role that controllability plays in the complexity of optimizing quantum control dynamics. We show that the loss of controllability generally leads to multiple locally suboptimal controls when gate fidelity in a quantum control system is maximized, which does not happen if the system is controllable. Such local suboptimal controls may attract an optimization algorithm into a local trap when a global optimal solution is sought, even if the target gate can be perfectly realized. This conclusion results from an analysis of the critical topology of the corresponding quantum control landscape, which refers to the gate fidelity objective as a functional of the control fields. For uncontrollable systems, due to SU(2) and SU(3) dynamical symmetries, the control landscape corresponding to an implementable target gate is proven to possess multiple locally optimal critical points, and its ruggedness can be further increased if the target gate is not realizable. These results imply that the optimization of quantum dynamics can be seriously impeded when operating with local search algorithms under these conditions, and thus full controllability is demanded. © 2011 American Physical Society.

Wu R.-B.,Tsinghua University | Wu R.-B.,Center for Quantum Information Science and Technology | Rabitz H.,Princeton University
Journal of Physics A: Mathematical and Theoretical | Year: 2012

The reliable realization of control operations is a key component of quantum information applications. In practice, meeting this goal is very demanding for open quantum systems. This paper investigates the landscape defined as the fidelity J between the desired and achieved quantum operations with an open system. The goal is to maximize J as a functional of the control variables. We specify the complete set of critical points of the landscape function in the so-called kinematic picture. An associated Hessian analysis of the landscape reveals that, upon the satisfaction of a particular controllability criterion, the critical topology is dependent on the particular environment, but no false traps (i.e. suboptimal solutions) exist. Thus, a gradient-type search algorithm should not be hindered in searching for the ultimate optimal solution with such controllable systems. Moreover, the maximal fidelity is proven to coincide with Uhlmann's fidelity between the environmental initial states associated with the achieved and desired quantum operations, which provides a generalization of Uhlmann's theorem in terms of Kraus maps. © 2012 IOP Publishing Ltd.

News Article
Site: www.rdmag.com

From gene mapping to space exploration, humanity continues to generate ever-larger sets of data -- far more information than people can actually process, manage, or understand. Machine learning systems can help researchers deal with this ever-growing flood of information. Some of the most powerful of these analytical tools are based on a strange branch of geometry called topology, which deals with properties that stay the same even when something is bent and stretched every which way. Such topological systems are especially useful for analyzing the connections in complex networks, such as the internal wiring of the brain, the U.S. power grid, or the global interconnections of the Internet. But even with the most powerful modern supercomputers, such problems remain daunting and impractical to solve. Now, a new approach that would use quantum computers to streamline these problems has been developed by researchers at MIT, the University of Waterloo, and the University of Southern California. The team describes their theoretical proposal this week in the journal Nature Communications. Seth Lloyd, the paper's lead author and the Nam P. Suh Professor of Mechanical Engineering, explains that algebraic topology is key to the new method. This approach, he says, helps to reduce the impact of the inevitable distortions that arise every time someone collects data about the real world. In a topological description, basic features of the data (How many holes does it have? How are the different parts connected?) are considered the same no matter how much they are stretched, compressed, or distorted. Lloyd explains that it is often these fundamental topological attributes "that are important in trying to reconstruct the underlying patterns in the real world that the data are supposed to represent." It doesn't matter what kind of dataset is being analyzed, he says. The topological approach to looking for connections and holes "works whether it's an actual physical hole, or the data represents a logical argument and there's a hole in the argument. This will find both kinds of holes." Using conventional computers, that approach is too demanding for all but the simplest situations. Topological analysis "represents a crucial way of getting at the significant features of the data, but it's computationally very expensive," Lloyd says. "This is where quantum mechanics kicks in." The new quantum-based approach, he says, could exponentially speed up such calculations. Lloyd offers an example to illustrate that potential speedup: If you have a dataset with 300 points, a conventional approach to analyzing all the topological features in that system would require "a computer the size of the universe," he says. That is, it would take 2300 (two to the 300th power) processing units -- approximately the number of all the particles in the universe. In other words, the problem is simply not solvable in that way. "That's where our algorithm kicks in," he says. Solving the same problem with the new system, using a quantum computer, would require just 300 quantum bits -- and a device this size may be achieved in the next few years, according to Lloyd. "Our algorithm shows that you don't need a big quantum computer to kick some serious topological butt," he says. There are many important kinds of huge datasets where the quantum-topological approach could be useful, Lloyd says, for example understanding interconnections in the brain. "By applying topological analysis to datasets gleaned by electroencephalography or functional MRI, you can reveal the complex connectivity and topology of the sequences of firing neurons that underlie our thought processes," he says. The same approach could be used for analyzing many other kinds of information. "You could apply it to the world's economy, or to social networks, or almost any system that involves long-range transport of goods or information," Lloyd says. But the limits of classical computation have prevented such approaches from being applied before. While this work is theoretical, "experimentalists have already contacted us about trying prototypes," he says. "You could find the topology of simple structures on a very simple quantum computer. People are trying proof-of-concept experiments." The team also included Silvano Garnerone of the University of Waterloo in Ontario, Canada, and Paolo Zanardi of the Center for Quantum Information Science and Technology at the University of Southern California.

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