arXiv subjects

Herschel Rabitz

Publications and source records attributed to Herschel Rabitz.

At least 37 records · Page 2Linked to original sources

arXiv2021

The Jacobi Theta Distribution

We form the Jacobi theta distribution through discrete integration of exponential random variables over an infinite inverse square law surface. It is continuous, supported on the positive reals, has a single positive parameter, is unimodal, positively skewed, and leptokurtic. Its cumulative distribution and density functions are expressed in terms of the Jacobi theta function. We describe asymptotic and log-normal approximations, inference, and a few applications of such distributions to modeling.

math.PR↗

arXiv2021

Positionally precise functionalization of shallow luminescent centers through Forster Resonant Energy Transfer (FRET) driven surface photochemistry

Paramagnetic luminescent impurities in solids such as Nitrogen-Vacancy (NV) centers in diamond represent a promising and versatile platform for the development of a wide range of chemical and biological sensors. This goal can be accomplished by the placement of chemically active, spin (electron or nuclear) labeled moiety on the surface in the close vicinity of a shallow single paramagnetic center. In this paper, we demonstrate that the Forster Resonant Excitation Transfer driven process where luminescent center plays a role of excitation donor can accomplish such a goal with high chemical efficiency, positionally precise and a scalable manner. We obtain the probability distribution function of the sensing group position relative to the luminescent center (NV center) and demonstrate that the functionalization position uncertainty is equal to the luminescent center deposition depth. The efficiency of the FRET process is analyzed as a function of the luminescent center deposition depth and the density of the excitation acceptor sites. Employing the geometric information of the surface-functionalized sensing groups, we obtain the probability distribution function of the energy gap in the spectrum of a spin-based detection system. This information allows us to estimate the single-molecule detection capabilities of the proposed system.

physics.app-ph↗

arXiv2021

Assessing three closed-loop learning algorithms by searching for high-quality quantum control pulses

Designing a high-quality control is crucial for reliable quantum computation. Among the existing approaches, closed-loop leaning control is an effective choice. Its efficiency depends on the learning algorithm employed, thus deserving algorithmic comparisons for its practical applications. Here, we assess three representative learning algorithms, including GRadient Ascent Pulse Engineering (GRAPE), improved Nelder-Mead (NMplus) and Differential Evolution (DE), by searching for high-quality control pulses to prepare the Bell state. We first implement each algorithm experimentally in a nuclear magnetic resonance system and then conduct a numerical study considering the impact of some possible significant experimental uncertainties. The experiments report the successful preparation of the high-fidelity target state with different convergence speeds by the three algorithms, and these results coincide with the numerical simulations when potential uncertainties are negligible. However, under certain significant uncertainties, these algorithms possess distinct performance with respect to their resulting precision and efficiency. This study provides insight to aid in the practical application of different closed-loop learning algorithms in realistic physical scenarios.

quant-ph↗

arXiv2020

Uncertainty Quantification by Random Measures and Fields

We present a general framework for uncertainty quantification that is a mosaic of interconnected models. We define global first and second order structural and correlative sensitivity analyses for random counting measures acting on risk functionals of input-output maps. These are the ANOVA decomposition of the intensity measure and the decomposition of the random measure variance, each into subspaces. Orthogonal random measures furnish sensitivity distributions. We show that the random counting measure may be used to construct positive random fields, which admit decompositions of covariance and sensitivity indices and may be used to represent interacting particle systems. The first and second order global sensitivity analyses conveyed through random counting measures elucidate and integrate different notions of uncertainty quantification, and the global sensitivity analysis of random fields conveys the proportionate functional contributions to covariance. This framework complements others when used in conjunction with for instance algorithmic uncertainty and model selection uncertainty frameworks.

math.PR↗

arXiv2020

Drawing together control landscape and tomography principles

The ability to control quantum systems using shaped fields as well as to infer the states of such controlled systems from measurement data are key tasks in the design and operation of quantum devices. Here we associate the success of performing both tasks to the structure of the underlying control landscape. We relate the ability to control and reconstruct the full state of the system to the absence of singular controls, and show that for sufficiently long evolution times singular controls rarely occur. Based on these findings, we describe a learning algorithm for finding optimal controls that makes use of measurement data obtained from partially accessing the system. Open challenges stemming from the concentration of measure phenomenon in high dimensional systems are discussed.

quant-ph↗

arXiv2020

Multi-Level Evolution Strategies for High-Resolution Black-Box Control

This paper introduces a multi-level (m-lev) mechanism into Evolution Strategies (ESs) in order to address a class of global optimization problems that could benefit from fine discretization of their decision variables. Such problems arise in engineering and scientific applications, which possess a multi-resolution control nature, and thus may be formulated either by means of low-resolution variants (providing coarser approximations with presumably lower accuracy for the general problem) or by high-resolution controls. A particular scientific application concerns practical Quantum Control (QC) problems, whose targeted optimal controls may be discretized to increasingly higher resolution, which in turn carries the potential to obtain better control yields. However, state-of-the-art derivative-free optimization heuristics for high-resolution formulations nominally call for an impractically large number of objective function calls. Therefore, an effective algorithmic treatment for such problems is needed. We introduce a framework with an automated scheme to facilitate guided-search over increasingly finer levels of control resolution for the optimization problem, whose on-the-fly learned parameters require careful adaptation. We instantiate the proposed m-lev self-adaptive ES framework by two specific strategies, namely the classical elitist single-child (1+1)-ES and the non-elitist multi-child derandomized $(μ_W,λ)$-sep-CMA-ES. We first show that the approach is suitable by simulation-based optimization of QC systems which were heretofore viewed as too complex to address. We also present a laboratory proof-of-concept for the proposed approach on a basic experimental QC system objective.

cs.NE↗

arXiv2020

Learning-based Quantum Robust Control: Algorithm, Applications and Experiments

Robust control design for quantum systems has been recognized as a key task in quantum information technology, molecular chemistry and atomic physics. In this paper, an improved differential evolution algorithm, referred to as \emph{msMS}\_DE, is proposed to search robust fields for various quantum control problems. In \emph{msMS}\_DE, multiple samples are used for fitness evaluation and a mixed strategy is employed for the mutation operation. In particular, the \emph{msMS}\_DE algorithm is applied to the control problems of (i) open inhomogeneous quantum ensembles and (ii) the consensus goal of a quantum network with uncertainties. Numerical results are presented to demonstrate the excellent performance of the improved machine learning algorithm for these two classes of quantum robust control problems. Furthermore, \emph{msMS}\_DE is experimentally implemented on femtosecond laser control applications to optimize two-photon absorption and control fragmentation of the molecule $\text{CH}_2\text{BrI}$. Experimental results demonstrate excellent performance of \emph{msMS}\_DE in searching for effective femtosecond laser pulses for various tasks.

quant-ph↗

arXiv2020

Quantum System Compression: A Hamiltonian Guided Walk Through Hilbert Space

We present a systematic study of quantum system compression for the evolution of generic many-body problems. The necessary numerical simulations of such systems are seriously hindered by the exponential growth of the Hilbert space dimension with the number of particles. For a \emph{constant} Hamiltonian system of Hilbert space dimension $n$ whose frequencies range from $f_{\min}$ to $f_{\max}$, we show via a proper orthogonal decomposition, that for a run-time $T$, the dominant dynamics are compressed in the neighborhood of a subspace whose dimension is the smallest integer larger than the time-bandwidth product $\delf=(f_{\max}-f_{\min})T$. We also show how the distribution of initial states can further compress the system dimension. Under the stated conditions, the time-bandwidth estimate reveals the \emph{existence} of an effective compressed model whose dimension is derived solely from system properties and not dependent on the particular implementation of a variational simulator, such as a machine learning system, or quantum device. However, finding an efficient solution procedure \emph{is} dependent on the simulator implementation{\color{black}, which is not discussed in this paper}. In addition, we show that the compression rendered by the proper orthogonal decomposition encoding method can be further strengthened via a multi-layer autoencoder. Finally, we present numerical illustrations to affirm the compression behavior in time-varying Hamiltonian dynamics in the presence of external fields. We also discuss the potential implications of the findings for machine learning tools to efficiently solve the many-body or other high dimensional Schr{ö}dinger equations.

quant-ph↗

arXiv2020

Learning control of quantum systems using frequency-domain optimization algorithms

We investigate two classes of quantum control problems by using frequency-domain optimization algorithms in the context of ultrafast laser control of quantum systems. In the first class, the system model is known and a frequency-domain gradient-based optimization algorithm is applied to searching for an optimal control field to selectively and robustly manipulate the population transfer in atomic Rubidium. The other class of quantum control problems involves an experimental system with an unknown model. In the case, we introduce a differential evolution algorithm with a mixed strategy to search for optimal control fields and demonstrate the capability in an ultrafast laser control experiment for the fragmentation of Pr(hfac)$_3$ molecules.

quant-ph↗

arXiv2020

Amplification of quadratic Hamiltonians

Speeding up the dynamics of a quantum system is of paramount importance for quantum technologies. However, in finite dimensions and without full knowledge of the details of the system, it is easily shown to be impossible. In contrast we show that continuous variable systems described by a certain class of quadratic Hamiltonians can be sped up without such detailed knowledge. We call the resultant procedure Hamiltonian amplification (HA). The HA method relies on the application of local squeezing operations allowing for amplifying even unknown or noisy couplings and frequencies by acting on individual modes. Furthermore, we show how to combine HA with dynamical decoupling to achieve amplified Hamiltonians that are free from environmental noise. Finally, we illustrate a significant reduction in gate times of cavity resonator qubits as one potential use of HA.

quant-ph↗

arXiv2020

Combining the synergistic control capabilities of modelling and experiments: illustration of finding a minimum time quantum objective

A common way to manipulate a quantum system, for example spins or artificial atoms, is to use properly tailored control pulses. In order to accomplish quantum information tasks before coherence is lost, it is crucial to implement the control in the shortest possible time. Here we report the near time-optimal preparation of a Bell state with fidelity higher than $99\%$ in an NMR experiment, which is feasible by combining the synergistic capabilities of modelling and experiments operating in tandem. The pulses preparing the Bell state are found by experiments that are recursively assisted with a gradient-based optimization algorithm working with a model. Thus, we explore the interplay between model-based numerical optimal design and experimental-based learning control. Utilizing the balanced synergism between the dual approaches should have broad applications for accelerating the search for optimal quantum controls.

quant-ph↗

arXiv2020

An upper bound on the time required to implement unitary operations

We derive an upper bound for the time needed to implement a generic unitary transformation in a $d$ dimensional quantum system using $d$ control fields. We show that given the ability to control the diagonal elements of the Hamiltonian, which allows for implementing any unitary transformation under the premise of controllability, the time $T$ needed is upper bounded by $T\leq \frac{πd^{2}(d-1)}{2g_{\text{min}}}$ where $g_{\text{min}}$ is the smallest coupling constant present in the system. We study the tightness of the bound by numerically investigating randomly generated systems, with specific focus on a system consisting of $d$ energy levels that interact in a tight-binding like manner.

quant-ph↗

arXiv2019

Robust Quantum Control in Games: an Adversarial Learning Approach

High-precision operation of quantum computing systems must be robust to uncertainties and noises in the quantum hardware. In this paper, we show that through a game played between the uncertainties (or noises) and the controls, adversarial uncertainty samples can be generated to find highly robust controls through the search for Nash equilibria (NE). We propose a broad family of adversarial learning algorithms, namely a-GRAPE algorithms, which include two effective learning schemes referred to as the best-response approach and the better-response approach within the game-theoretic terminology, providing options for rapidly learning robust controls. Numerical experiments demonstrate that the balance between fidelity and robustness depends on the details of the chosen adversarial learning algorithm, which can effectively lead to a significant enhancement of control robustness while attaining high fidelity.

quant-ph↗

arXiv2019

Numerical meshless solution of high-dimensional sine-Gordon equations via Fourier HDMR-HC approximation

In this paper, an implicit time stepping meshless scheme is proposed to find the numerical solution of high-dimensional sine-Gordon equations (SGEs) by combining the high dimensional model representation (HDMR) and the Fourier hyperbolic cross (HC) approximation. To ensure the sparseness of the relevant coefficient matrices of the implicit time stepping scheme, the whole domain is first divided into a set of subdomains, and the relevant derivatives in high-dimension can be separately approximated by the Fourier HDMR-HC approximation in each subdomain. The proposed method allows for stable large time-steps and a relatively small number of nodes with satisfactory accuracy. The numerical examples show that the proposed method is very attractive for simulating the high-dimensional SGEs.

math.NA↗

arXiv2019

Meshless Hermite-HDMR finite difference method for high-dimensional Dirichlet problems

In this paper, a meshless Hermite-HDMR finite difference method is proposed to solve high-dimensional Dirichlet problems. The approach is based on the local Hermite-HDMR expansion with an additional smoothing technique. First, we introduce the HDMR decomposition combined with the multiple Hermite series to construct a class of Hermite-HDMR approximations, and the relevant error estimate is theoretically built in a class of Hermite spaces. It can not only provide high order convergence but also retain good scaling with increasing dimensions. Then the Hermite-HDMR based finite difference method is particularly proposed for solving high-dimensional Dirichlet problems. By applying a smoothing process to the Hermite-HDMR approximations, numerical stability can be guaranteed even with a small number of nodes. Numerical experiments in dimensions up to $30$ show that resulting approximations are of very high quality.

math.NA↗

arXiv2018

Optimal control of interacting quantum systems based on the first-order Magnus approximation: Application to multiple dipole-dipole coupled molecular rotors

We develop a methodology for performing approximate optimal control simulations for quantum systems with multiple interacting degrees of freedom. The quantum dynamics are modeled using the first-order Magnus approximation in the interaction picture, where the interactions between different degrees of freedom are treated as the perturbation. We present a numerical procedure for implementing this approximation, which leverages the separability of the zeroth-order time evolution operator and the pairwise nature of common interactions for a reduced computational cost. This formulation of the first-order Magnus approximation is suitable to be combined with gradient-free methods for control field optimization; to this end, we adopt a Stochastic Hill Climbing algorithm. The associated computational costs are analyzed and compared with those of the exact simulation in the large N limit. For numerical illustrations, we perform approximate optimal control simulations for systems of two and three dipole-dipole coupled molecular rotors under the influence of a global control field. For the two rotor system, we optimize fields for both orientation and entanglement control objectives. For the three rotor system, we optimize fields for orienting rotors in the same direction as well as in opposite directions.

quant-ph↗

arXiv2018

Suppression of the Spectral Cross Talk of Optogenetic Switching by Stimulated Depletion Quenching. Theoretical Analysis

Optogenetics is a rapidly growing field of biotechnology, potentially allowing a deeper understanding and control of complex biological networks. The major challenge is the multiplexed control of several optogenetic components in the presence of significant spectral cross talk. We propose and demonstrate through simulations a new control approach of Stimulated Depletion Quenching. This approach is applied to the phytochrome Cph8 bidirectional optogenetic switch, and the results show significant improvement of its dynamic range.

q-bio.QM↗

arXiv2018

Coherent Control of Optogenetic Switching by Stimulated Depletion Quenching

Optogenetics is a revolutionary new field of biotechnology, achieving optical control over biological functions in living cells by genetically inserting light sensitive proteins into cellular signaling pathways. Applications of optogenetic switches are expanding rapidly, but the technique is hampered by spectral cross-talk: the broad absorption spectra of compatible biochemical chromophores limits the number of switches that can be independently controlled and restricts the dynamic range of each switch. In the present work we develop and implement a non-linear optical photoswitching capability, Stimulated Depletion Quenching (SDQ), is used to overcome spectral cross-talk by exploiting the molecules' unique dynamic response to ultrashort laser pulses. SDQ is employed to enhance the control of Cph8, a photo-reversible phytochrome based optogenetic switch designed to control gene expression in E. Coli bacteria. The Cph8 switch can not be fully converted to it's biologically inactive state ($P_{FR}$) by linear photos-witching, as spectral cross-talk causes a reverse photoswitching reaction to revert to it back to the active state ($P_{R}$). SDQ selectively halts this reverse reaction while allowing the forward reaction to proceed. The results of this proof of concept experiment lay the foundation for future experiments that will use optimal pulse shaping to further enhance control of Cph8 and enable simultaneous, multiplexed control of multiple optogenetic switches.

q-bio.QM↗