arXiv subjects

Herschel Rabitz

Publications and source records attributed to Herschel Rabitz.

At least 55 records · Page 3Linked to original sources

arXiv2018

Quantum Control Landscape of Bipartite Systems

The control landscape of a quantum system $A$ interacting with another quantum system $B$ is studied. Only system $A$ is accessible through time dependent controls, while system B is not accessible. The objective is to find controls that implement a desired unitary transformation on $A$, regardless of the evolution on $B$, at a sufficiently large final time. The freedom in the evolution on $B$ is used to define an \emph{extended control landscape} on which the critical points are investigated in terms of kinematic and dynamic gradients. A spectral decomposition of the corresponding extended unitary system simplifies the landscape analysis which provides: (i) a sufficient condition on the rank of the dynamic gradient of the extended landscape that guarantees a trap free search for the final time unitary matrix of system $A$, and (ii) a detailed decomposition of the components of the overall dynamic gradient matrix. Consequently, if the rank condition is satisfied, a gradient algorithm will find the controls that implements the target unitary on system $A$. It is shown that even if the dynamic gradient with respect to the controls alone is not full rank, the additional flexibility due to the parameters that define the extended landscape still can allow for the rank condition of the extended landscape to hold. Moreover, satisfaction of the latter rank condition subsumes any assumptions about controllability, reachability and control resources. Here satisfaction of the rank condition is taken as an assumption. The conditions which ensure that it holds remain an open research question. We lend some numerical support with two common examples for which the rank condition holds.

quant-ph↗

arXiv2018

Singularity-free quantum tracking control of molecular rotor orientation

Quantum tracking control aims to identify applied fields to steer the expectation values of particular observables along desired paths in time. The associated temporal fields can be identified by inverting the underlying dynamical equations for the observables. However, fields found in this manner are often plagued by undesirable singularities. In this paper we consider a planar molecular rotor, and derive singularity-free tracking expressions for the fields that steer the expectation of the orientation of the rotor along desired trajectories in time. Simulations are presented that utilize two orthogonal control electric fields to drive the orientation of the rotor along a series of designated tracks.

quant-ph↗

arXiv2018

Targeted Nonlinear Adversarial Perturbations in Images and Videos

We introduce a method for learning adversarial perturbations targeted to individual images or videos. The learned perturbations are found to be sparse while at the same time containing a high level of feature detail. Thus, the extracted perturbations allow a form of object or action recognition and provide insights into what features the studied deep neural network models consider important when reaching their classification decisions. From an adversarial point of view, the sparse perturbations successfully confused the models into misclassifying, although the perturbed samples still belonged to the same original class by visual examination. This is discussed in terms of a prospective data augmentation scheme. The sparse yet high-quality perturbations may also be leveraged for image or video compression.

cs.CV↗

arXiv2018

High Dimensional Model Representation as a Glass Box in Supervised Machine Learning

Prediction and explanation are key objects in supervised machine learning, where predictive models are known as black boxes and explanatory models are known as glass boxes. Explanation provides the necessary and sufficient information to interpret the model output in terms of the model input. It includes assessments of model output dependence on important input variables and measures of input variable importance to model output. High dimensional model representation (HDMR), also known as the generalized functional ANOVA expansion, provides useful insight into the input-output behavior of supervised machine learning models. This article gives applications of HDMR in supervised machine learning. The first application is characterizing information leakage in ``big-data'' settings. The second application is reduced-order representation of elementary symmetric polynomials. The third application is analysis of variance with correlated variables. The last application is estimation of HDMR from kernel machine and decision tree black box representations. These results suggest HDMR to have broad utility within machine learning as a glass box representation.

stat.ME↗

arXiv2018

Controlling qubit networks in polynomial time

Future quantum devices often rely on favourable scaling with respect to the system components. To achieve desirable scaling, it is therefore crucial to implement unitary transformations in an efficient manner. We develop an upper bound for the minimum time required to implement a unitary transformation on a generic qubit network in which each of the qubits is subject to local time dependent controls. The set of gates is characterized that can be implemented in a time that scales at most polynomially in the number of qubits. Furthermore, we show how qubit systems can be concatenated through controllable two body interactions, making it possible to implement the gate set efficiently on the combined system. Finally a system is identified for which the gate set can be implemented with fewer controls. The considered model is particularly important, since it describes electron-nuclear spin interactions in NV centers.

quant-ph↗

arXiv2018

A Data-Driven Gradient Algorithm for High-Precision Quantum Control

In the quest to achieve scalable quantum information processing technologies, gradient-based optimal control algorithms (e.g., GRAPE) are broadly used for implementing high-precision quantum gates, but their performance is often hindered by deterministic or random errors in the system model and the control electronics. In this paper, we show that GRAPE can be taught to be more effective by jointly learning from the design model and the experimental data obtained from process tomography. The resulting data-driven gradient optimization algorithm (d-GRAPE) can in principle correct all deterministic gate errors, with a mild efficiency loss. The d-GRAPE algorithm may become more powerful with broadband controls that involve a large number of control parameters, while other algorithms usually slow down due to the increased size of the search space. These advantages are demonstrated by simulating the implementation of a two-qubit CNOT gate.

quant-ph↗

arXiv2018

Dependence of the Quantum Speed Limit on System Size and Control Complexity

We extend the work in New J. Phys. 19, 103015 (2017) by deriving a lower bound for the minimum time necessary to implement a unitary transformation on a generic, closed quantum system with an arbitrary number of classical control fields. This bound is explicitly analyzed for a specific N-level system similar to those used to represent simple models of an atom, or the first excitation sector of a Heisenberg spin chain, both of which are of interest in quantum control for quantum computation. Specifically, it is shown that the resultant bound depends on the dimension of the system, and on the number of controls used to implement a specific target unitary operation. The value of the bound determined numerically, and an estimate of the true minimum gate time are systematically compared for a range of system dimension and number of controls; special attention is drawn to the relationship between these two variables. It is seen that the bound captures the scaling of the minimum time well for the systems studied, and quantitatively is correct in the order of magnitude.

quant-ph↗

arXiv2017

Pulse Width Modulation method for quantum control design

Illuminated by the Pulse Width Modulation (PWM) technology in classical control engineering, we propose the PWM approximation which transforms continuous and bang-bang control into each other. This method works by squeezing the approximation error into high-frequency components above a prescribed cutoff frequency $Ω$, which can be tailored by modulating the switch time of the bang-bang control. Because the quantum dynamics under bang-bang control can be efficiently calculated, PWM allows us to speed up quantum control design particularly in large dimensional systems with small number of control variables. Moreover, PWM paves the way to implement various bang-bang control protocols in the literature with an arbitrary waveform, which is more desirable in laboratory.

quant-ph↗

arXiv2017

The roles of drift and control field constraints upon quantum control speed limits

In this work we derive a lower bound for the minimum time required to implement a target unitary transformation through a classical time-dependent field in a closed quantum system. The bound depends on the target gate, the strength of the internal Hamiltonian and the highest permitted control field amplitude. These findings reveal some properties of the reachable set of operations, explicitly analyzed for a single qubit. Moreover, for fully controllable systems, we identify a lower bound for the time at which all unitary gates become reachable. We use numerical gate optimization in order to study the tightness of the obtained bounds. It is shown that in the single qubit case our analytical findings describe the relationship between the highest control field amplitude and the minimum evolution time remarkably well. Finally, we discuss both challenges and ways forward for obtaining tighter bounds for higher dimensional systems, offering a discussion about the mathematical form and the physical meaning of the bound.

quant-ph↗

arXiv2017

Control landscapes for a class of non-linear dynamical systems: sufficient conditions for the absence of traps

We establish three tractable, jointly sufficient conditions for the control landscapes of non-linear control systems to be trap free comparable to those now well known in quantum control. In particular, our results encompass end-point control problems for a general class of non-linear control systems of the form of a linear time invariant term with an additional state dependent non-linear term. Trap free landscapes ensure that local optimization methods (such as gradient ascent) can achieve monotonic convergence to effective control schemes in both simulation and practice. Within a large class of non-linear control problems, each of the three conditions is shown to hold for all but a null set of cases. Furthermore, we establish a Lipschitz condition for two of these assumptions; under specific circumstances, we explicitly find the associated Lipschitz constants. A detailed numerical investigation using the D-MOPRH control optimization algorithm is presented for a specific family of systems which meet the conditions for possessing trap free control landscapes. The results obtained confirm the trap free nature of the landscapes of such systems.

math.OC↗

arXiv2017

Making Distinct Dynamical Systems Appear Spectrally Identical

We show that a laser pulse can always be found that induces a desired optical response from an arbitrary dynamical system. As illustrations, driving fields are computed to induce the same optical response from a variety of distinct systems (open and closed, quantum and classical). As a result, the observed induced dipolar spectra without detailed information on the driving field is not sufficient to characterize atomic and molecular systems. The formulation may also be applied to design materials with specified optical characteristics. These findings reveal unexplored flexibilities of nonlinear optics.

quant-ph↗

arXiv2016

Searching for an optimal control in the presence of saddles on the quantum mechanical observable landscape

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 the associated landscape also contains saddles, i.e., critical points that are not local suboptimal extrema. In this paper, we investigate whether the presence of these saddles affects the trajectories of gradient-based searches for an optimal control. We show through simulations that both the detailed topology of the control landscape and the parameters of the system Hamiltonian influence whether the searches are attracted to a saddle. For some circumstances with a special initial state and target observable, optimizations may approach a saddle very closely, reducing the efficiency of the gradient algorithm. Encounters with such attractive saddles are found to be quite rare. Neither the presence of a large number of saddles on the control landscape nor a large number of system states increase the likelihood that a search will closely approach a saddle. Even for applications that encounter a saddle, well-designed gradient searches with carefully chosen algorithmic parameters will readily locate optimal controls.

quant-ph↗

arXiv2016

Inherently trap-free convex landscapes for full quantum optimal control

We present a comprehensive analysis of the landscape for full quantum-quantum control associated with the expectation value of an arbitrary observable of one quantum system controlled by another quantum system. It is shown that such full quantum-quantum control landscapes are convex, and hence devoid of local suboptima and saddle points that may exist in landscapes for quantum systems controlled by time-dependent classical fields. There is no controllability requirement for the full quantum-quantum landscape to be trap-free, although the forms of Hamiltonians, the flexibility in choosing initial state of the controller, as well as the control duration, can infulence the reachable optimal value on the landscape. All level sets of the full quantum-quantum landscape are connected convex sets. Finally, we show that the optimal solution of the full quantum-quantum control landscape can be readily determined numerically, which is demonstrated using the Jaynes-Cummings model depicting a two-level atom interacting with a quantized radiation field.

quant-ph↗

arXiv2016

Common Foundations of Optimal Control Across the Sciences: evidence of a free lunch

A common goal in the sciences is optimization of an objective function by selecting control variables such that a desired outcome is achieved. This scenario can be expressed in terms of a control landscape of an objective considered as a function of the control variables. At the most basic level, it is known that the vast majority of quantum control landscapes possess no traps, whose presence would hinder reaching the objective. This paper reviews and extends the quantum control landscape assessment, presenting evidence that the same highly favorable landscape features exist in many other domains of science. The implications of this broader evidence are discussed. Specifically, control landscape examples from quantum mechanics, chemistry, and evolutionary biology are presented. Despite the obvious differences, commonalities between these areas are highlighted within a unified mathematical framework. This mathematical framework is driven by the wide ranging experimental evidence on the ease of finding optimal controls (in terms of the required algorithmic search effort beyond laboratory set up overhead). The full scope and implications of this observed common control behavior pose an open question for assessment in further work.

quant-ph↗

arXiv2016

Quantum Control Landscapes Are Almost Always Trap Free

A proof that almost all quantum systems have trap free (that is, free from local optima) landscapes is presented for a large and physically general class of quantum system. This result offers an explanation for why gradient methods succeed so frequently in quantum control in both theory and practice. The role of singular controls is analyzed using geometric tools in the case of the control of the propagator of closed finite dimension systems. This type of control field has been implicated as a source of landscape traps. The conditions under which singular controls can introduce traps, and thus interrupt the progress of a control optimization, are discussed and a geometrical characterization of the issue is presented. It is shown that a control being singular is not sufficient to cause a control optimization progress to halt and sufficient conditions for a trap free landscape are presented. It is further shown that the local surjectivity axiom of landscape analysis can be refined to the condition that the end-point map is transverse to each of the level sets of the fidelity function. This novel condition is shown to be sufficient for a quantum system's landscape to be trap free. The control landscape for a quantum system is shown to be trap free for all but a null set of Hamiltonians using a novel geometric technique based on the parametric transversality theorem. Numerical evidence confirming this is also presented. This result is the analogue of the work of Altifini, wherein it is shown that controllability holds for all but a null set of quantum systems in the dipole approximation. The presented results indicate that by-and-large limited control resources are the most physically relevant source of landscape traps.

quant-ph↗

arXiv2016

Fast Quantum Communication in Linear Networks

Here we consider the speed at which quantum information can be transferred between the nodes of a linear network. Because such nodes are linear oscillators, this speed is also important in the cooling and state preparation of mechanical oscillators, as well as frequency conversion. We show that if there is no restriction on the size of the linear coupling between two oscillators, then there exist control protocols that will swap their respective states with high fidelity within a time much less than a single oscillation period. Standard gradient search methods fail to find these fast protocols. We were able to do so by augmenting standard search methods with a path-tracing technique, demonstrating that this technique has remarkable power to solve time-optimal control problems, as well as confirming the highly challenging nature of these problems. As a further demonstration of the power of path-tracing, first introduced by Moore-Tibbets et al. [Phys. Rev. A 86, 062309 (2012)], we apply it to the generation of entanglement in a linear network.

quant-ph↗

arXiv2016

Optimal nonlinear coherent mode transitions in Bose-Einstein Condensates utilizing spatio-temporal controls

Bose-Einstein condensates (BECs) offer the potential to examine quantum behavior at large length and time scales, as well as forming promising candidates for quantum technology applications. Thus, the manipulation of BECs using control fields is a topic of prime interest. We consider BECs in the mean field model of the Gross-Pitaevskii equation (GPE), which contains linear and nonlinear features, both of which are subject to control. In this work we report successful optimal control simulations of a one dimensional GPE by modulating the linear and nonlinear terms to stimulate transitions into excited coherent modes. The linear and nonlinear controls are allowed to freely vary over space and time to seek their optimal forms. The determination of the excited coherent modes targeted for optimization is numerically performed through an adaptive imaginary time propagation method. Numerical simulations are performed for optimal control of mode-to-mode transitions between the ground coherent mode and excited modes of a BEC trapped in a harmonic well. The results show greater than 99% success for nearly all trials utilizing reasonable initial guesses for the controls, and analysis of the optimal controls reveals primarily direct transitions between initial and target modes. The success of using solely the nonlinearity term as a control opens up further research toward exploring novel control mechanisms inaccessible to linear Schrödinger-type systems.

quant-ph↗

arXiv2016

Quantum Control Landscapes Beyond the Dipole Approximation

We investigate the control landscapes of closed, finite level quantum systems beyond the dipole approximation by including a polarizability term in the Hamiltonian. Theoretical analysis is presented for the $n$ level case and formulas for singular controls, which are candidates for landscape traps, are compared to their analogues in the dipole approximation. A numerical analysis of the existence of traps in control landscapes beyond the dipole approximation is made in the four level case. A numerical exploration of these control landscapes is achieved by generating many random Hamiltonians which include a term quadratic in a single control field. The landscapes of such systems are found numerically to be trap free in general. This extends a great body of recent work on typical landscapes of quantum systems where the dipole approximation is made. We further investigate the relationship between the magnitude of the polarizability and the magnitude of the controls resulting from optimization. It is shown numerically that including a polarizability term in an otherwise uncontrollable system removes traps from the landscapes of a specific family of systems by restoring controllability. We numerically assess the effect of a random polarizability term on the know example of a three level system with a second order trap in its control landscape. It is found that the addition of polarizability removes the trap from the landscape. The implications for laboratory control are discussed.

quant-ph↗