Evolution Operator Research Articles

Walking on vertices and edges by continuous-time quantum walk

The quantum walk dynamics obey the laws of quantum mechanics with an extra locality constraint, which demands that the evolution operator is local in the sense that the walker must visit the neighboring locations before endeavoring to distant places. Usually, the Hamiltonian is obtained from either the adjacency or the Laplacian matrix of the graph and the walker hops from vertices to neighboring vertices. In this work, we define a version of the continuous-time quantum walk that allows the walker to hop from vertices to edges and vice versa. As an application, we analyze the spatial search algorithm on the complete bipartite graph by modifying the new version of the Hamiltonian with an extra term that depends on the location of the marked vertex or marked edge, similar to what is done in the standard continuous-time quantum walk model. We show that the optimal running time to find either a vertex or an edge is \(O(\sqrt{N_\textrm{e}})\) with success probability \(1-o(1)\), where \(N_\textrm{e}\) is the number of edges of the complete bipartite graph.

Low-Depth Hamiltonian Simulation by an Adaptive Product Formula.

Various Hamiltonian simulation algorithms have been proposed to efficiently study the dynamics of quantum systems on a quantum computer. The existing algorithms generally approximate the time evolution operators, which may need a deep quantum circuit that is beyond the capability of near-term noisy quantum devices. Here, focusing on the time evolution of a fixed input quantum state, we propose an adaptive approach to construct a low-depth time evolution circuit. By introducing a measurable quantifier that characterizes the simulation error, we use an adaptive strategy to learn the shallow quantum circuit that minimizes that error. We numerically test the adaptive method with electronic Hamiltonians of the H_{2}O and H_{4} molecules, and the transverse field Ising model with random coefficients. Compared to the first-order Suzuki-Trotter product formula, our method can significantly reduce the circuit depth (specifically the number of two-qubit gates) by around two orders while maintaining the simulation accuracy. We show applications of the method in simulating many-body dynamics and solving energy spectra with the quantum Krylov algorithm. Our work sheds light on practical Hamiltonian simulation with noisy-intermediate-scale-quantum devices.

Topological characterization and stability of Floquet Majorana modes in Rashba nanowires

We theoretically investigate a practically realizable Floquet topological superconductor model based on a one-dimensional Rashba nanowire and proximity-induced $s$-wave superconductivity in the presence of a Zeeman field. The driven system hosts regular 0-Majorana end modes and anomalous $\ensuremath{\pi}$-Majorana end modes (MEMs). By tuning the chemical potential and the frequency of the drive, we illustrate the generation of multiple MEMs in our theoretical setup. We utilize the chiral symmetry operator to topologically characterize these MEMs via a dynamical winding number constructed out of the periodized evolution operator. Interestingly, the robustness of the 0- and $\ensuremath{\pi}$-MEMs is established in the presence of on-site time-independent random disorder potential. We employ the twisted boundary condition to define the dynamical topological invariant for this translational-symmetry broken system. The interplay between the Floquet driving and the weak disorder can stabilize the MEMs, giving rise to a quantized value of the dynamical winding number for a finite range of drive parameters. This observation might be experimentally helpful in scrutinizing the topological nature of the Floquet MEMs. We showcase another driving protocol, namely, a periodic kick in the chemical potential, to study the generation of Floquet MEMs in our setup. Our work paves a realistic way to engineer multiple MEMs in a driven system.

One-dimensional transient crystals

A phase of solid state matter which exists in time dimension is investigated theoretically for a linear chain of quantum dots or atomic sites using the evolution operator technique and the Laplace transform technique.. The studies focus on the spectral density of states (DOS) function and its modifications after a sudden quench in the system. It is shown that this function reveals a very regular structure which oscillates in time and which corresponds to the stationary DOS of one-dimensional crystals. Thus, such a system stands for a transient crystal material which exists in time dimension but appears only for a short period and suddenly vanishes. Moreover, we observe different impulse propagation velocities due to the local DOS inertia, i.e., it is more difficult to rebuild a many-peaked structure of DOS related to longer chains. We also show that the transient crystal pattern in a linear chain can be observed for real electrodes characterized by the van Hove singularities or by a flat spectral density.

A graph-based framework for model-driven optimization facilitating impact analysis of mutation operator properties

Optimization problems in software engineering typically deal with structures as they occur in the design and maintenance of software systems. In model-driven optimization (MDO), domain-specific models are used to represent these structures while evolutionary algorithms are often used to solve optimization problems. However, designing appropriate models and evolutionary algorithms to represent and evolve structures is not always straightforward. Domain experts often need deep knowledge of how to configure an evolutionary algorithm. This makes the use of model-driven meta-heuristic search difficult and expensive. We present a graph-based framework for MDO that identifies and clarifies core concepts and relies on mutation operators to specify evolutionary change. This framework is intended to help domain experts develop and study evolutionary algorithms based on domain-specific models and operators. In addition, it can help in clarifying the critical factors for conducting reproducible experiments in MDO. Based on the framework, we are able to take a first step toward identifying and studying important properties of evolutionary operators in the context of MDO. As a showcase, we investigate the impact of soundness and completeness at the level of mutation operator sets on the effectiveness and efficiency of evolutionary algorithms.

Evolutionary competitive multitasking optimization via improved adaptive differential evolution

Competitive multitasking optimization (CMTO) is a special multitasking optimization paradigm that has been recently proposed. In the CMTO problems, the objective values of all tasks are competitive, and the purpose of CMTO is to find an optimal solution for multiple tasks. However, existing algorithms designed for the CMTO problems perform poorly since their resource allocation strategies are prone to incorrect task selection. To remedy this drawback, this paper proposes an improved multitasking adaptive differential evolution, which can be featured as: (i) a success-history based resource allocation strategy is proposed; (ii) an adaptive random mating probability control strategy is devised to adapt to different task combinations; (iii) an adaptive multitasking differential evolution operator is designed to enhance the searchability. To evaluate the performance of the proposed method, three CMTO benchmark test suites and two real-world optimization problems are chosen. Compared with other related methods, the results show that the proposed method achieved better performance empirically.

Memory, evolutionary operator, and local search based improved Grey Wolf Optimizer with linear population size reduction technique

Optimization of multi-modal functions is challenging even for evolutionary and swarm-based algorithms as it requires an efficient exploration for finding the promising region of the search space, and effective exploitation to precisely find the global optimum. Grey Wolf Optimizer (GWO) is a recently developed metaheuristic algorithm that is inspired by nature with a relatively small number of parameters for tuning. However, GWO and most of its variants may suffer from the lack of population diversity, premature convergence, and the inability to preserve a good balance between exploratory and exploitative behaviors. To address these limitations, this work proposes a new variant of GWO incorporating memory, evolutionary operators, and a stochastic local search technique. It further integrates Linear Population Size Reduction (LPSR) technique. The proposed algorithm is comprehensively tested on 23 numerical benchmark functions, high dimensional benchmark functions, 13 engineering case studies, four data classifications, and three function approximation problems. The benchmark functions are mostly taken from the CEC 2005 and CEC 2010 special sessions, and they include rotated, shifted functions. The engineering case studies are from the CEC 2020 real-world non-convex constrained optimization problems. The performance of the proposed GWO is compared with popular metaheuristics, namely, particle swarm optimization (PSO), gravitational search algorithm (GSA), slap swarm algorithm (SSA), differential evolution (DE), self-adaptive differential evolution (SADE), basic GWO and its three recently improved variants. Statistical analysis and Friedman tests have been conducted to thoroughly compare their performance. The obtained results demonstrate that the proposed GWO outperforms the algorithms compared for the benchmark functions and engineering case studies tested.

Non-Hermitian Hamiltonian deformations in quantum mechanics

The construction of exactly-solvable models has recently been advanced by considering integrable Toverline{T} deformations and related Hamiltonian deformations in quantum mechanics. We introduce a broader class of non-Hermitian Hamiltonian deformations in a nonrelativistic setting, to account for the description of a large class of open quantum systems, which includes, e.g., arbitrary Markovian evolutions conditioned to the absence of quantum jumps. We relate the time evolution operator and the time-evolving density matrix in the undeformed and deformed theories in terms of integral transforms with a specific kernel. Non-Hermitian Hamiltonian deformations naturally arise in the description of energy diffusion that emerges in quantum systems from time-keeping errors in a real clock used to track time evolution. We show that the latter can be related to an inverse Toverline{T} deformation with a purely imaginary deformation parameter. In this case, the integral transforms take a particularly simple form when the initial state is a coherent Gibbs state or a thermofield double state, as we illustrate by characterizing the purity, Rényi entropies, logarithmic negativity, and the spectral form factor. As the dissipative evolution of a quantum system can be conveniently described in Liouville space, we further study the spectral properties of the Liouvillians, i.e., the dynamical generators associated with the deformed theories. As an application, we discuss the interplay between decoherence and quantum chaos in non-Hermitian deformations of random matrix Hamiltonians and the Sachdev-Ye-Kitaev model.

An optimization numerical spiking neural P system for solving constrained optimization problems

An optimization spiking neural P (OSN P) system is a discrete optimization model without the aid of evolutionary operators of evolutionary algorithms or swarm intelligence algorithms. However, since the processing object of OSN P systems is a spike, where information is encoded by the timing of spikes or the number of spikes in neurons, OSN P systems are limited for solving continuous optimization problems. To break this limitation, an extended numerical spiking neural (ENSN P) system is proposed based on numerical spiking neural P (NSN P) systems and multiple (ENSN P) systems, called optimization numerical spiking neural P systems (ONSN P systems or ONSNPS), are designed to solve continuous constrained optimization problems. More specifically, in ENSN P systems, the production functions are selected by probability to achieve updated parameters. In OSN P systems, a guider algorithm is introduced to finish individuals’ crossover and selection. The extensively experimental results in five benchmarks, thirty-two optimization problems including five benchmark problems, seventeen manufacturing design optimization problems and ten benchmarks from CEC show that ONSN P systems in this paper outperform or are competitive to twenty-eight optimization algorithms. Finally, algorithm complexity and Holm-Bonferroni procedure based on statistical results is used to test the complexity changing when we use different dimensionality of the search space and the difference in terms of statistical performance. The testing results indicate that the time complexity of ONSN P systems grows linearly with problem dimensions and ONSN P systems are better performance than the most algorithms.

Unbounded generalization of logarithmic representation of infinitesimal generators

The logarithmic representation of infinitesimal generators is generalized to the cases when the evolution operator is unbounded. The generalized result is applicable to the representation of infinitesimal generators of unbounded evolution operators, where unboundedness of evolution operator is an essential ingredient of nonlinear analysis. In conclusion, a general framework for the identification between the infinitesimal generators with evolution operators is established. A mathematical framework for such an identification is indispensable to the rigorous treatment of nonlinear transforms, for example, transforms appearing in the theory of integrable systems.

Approximate Controllability of Non-autonomous Second Order Impulsive Functional Evolution Equations in Banach Spaces

This article investigates the approximate controllability of second order non-autonomous functional evolution equations involving non-instantaneous impulses and nonlocal conditions. First, we discuss the approximate controllability of second order linear system in detail, which lacks in the existing literature. Then, we derive sufficient conditions for approximate controllability of our system in separable reflexive Banach spaces via linear evolution operator, resolvent operator conditions, and Schauder’s fixed point theorem. Moreover, in this paper, we define proper identification of resolvent operator in Banach spaces. Finally, we provide two concrete examples to validate our results.

Reliability and efficiency enhancement of a radial distribution system through value-based auto-recloser placement and network remodelling

The electric distribution system (EDS) is prone to faults leading to power interruptions. The present energy market demands that electricity utilities invest more in different measures to improve the performance of the EDS. The approach proposed here details a composite dual-phased methodology to improve the reliability and efficiency of the power delivered by the EDS. In the first phase, the optimal allocation of auto-reclosers (AR) is undertaken by employing a newly formulated algorithm. The determination of the total number and location for AR placement is based on the economic analysis of two factors, i.e., AR investment-maintenance cost and total benefit earned in terms of reliability improvement due to AR placement. The analysis also takes into account the impact of power outages on different load types, the load growth rate, and the inflation rate. Further, to enhance the efficiency of the AR-incorporated EDS, the technique of Radial Distribution System Remodelling is employed in the second phase. This method searches for a radial configuration that delivers power at minimum line losses. These phases comprising complex combinatorial operations are aided by a fresh hybrid of the Sine Cosine Algorithm, Krill Herd Algorithm, and a genetic operator of Differential Evolution. The results obtained from its application on the IEEE 69-bus distribution test system prove the credibility of the suggested formulation.

Graphene-Based Derivatives Heterostructured Catalytic Systems for Sustainable Hydrogen Energy via Overall Water Splitting

The global climate crisis has cultivated the demand for sustainable energy resources as fossil derivative fuels are functional in catalyzing the rate of environmental breakdown. Sustainable energy solutions generate various renewable energy prospects capable of delivering efficient energy operations. Among these prospects, green H2 energy generated via overall water splitting is an effective approach towards sustainability ascribed to the higher gravimetric density and efficiency of H2 fuel. In this review, we sought to discuss the applicability and challenges of graphene-based derivatives in H2 evolution operations through photochemical, electrochemical and photoelectrochemical water-splitting pathways. The unique layered structure of graphene-based derivatives alongside marvelous optoelectronic and physicochemical properties ease out the thermodynamic uphill of water splitting better than their non-layered counterparts. In addition, the heterojunction formation in the graphene derivatives with visible light catalysts propels the kinetics of HER. Functionalized GO and rGO derivatives of graphene are riveting catalysts that have received extensive interest from researchers attributed to their accelerated chemical and mechanical stability, tunable band structure and larger surface area, providing more exposed active sites for HER. The surface organic functional groups of GO/rGO assist in establishing synergetic interfacial contact with other catalysts. Thus, these groups provide structural and chemical versatility to GO/rGO-based heterostructured catalysts, which effectively improve their physicochemical parameters that drive their catalytic performance towards HER. In order to develop a cost-effective and highly efficient catalytic system, graphene-based derivatives are promising heterostructured catalysts that exhibit a good relationship between catalytic efficiency and robustness.

Fundamental transfer matrix for electromagnetic waves, scattering by a planar collection of point scatterers, and anti- PT -symmetry

We develop a fundamental transfer-matrix formulation of the scattering of electromagnetic (EM) waves that incorporates the contribution of the evanescent waves and applies to general stationary linear media which need not be isotropic, homogenous, or passive. Unlike the traditional transfer matrices whose definition involves slicing the medium, the fundamental transfer matrix is a linear operator acting in an infinite-dimensional function space. It is given in terms of the evolution operator for a nonunitary quantum system and has the benefit of allowing for analytic calculations. In this respect it is the only available alternative to the standard Green's-function approaches to EM scattering. We use it to offer an exact solution of the outstanding EM scattering problem for an arbitrary finite collection of possibly anisotropic nonmagnetic point scatterers lying on a plane. In particular, we provide a comprehensive treatment of doublets consisting of pairs of isotropic point scatterers and study their spectral singularities. We show that identical and $\mathcal{P}\mathcal{T}$-symmetric doublets do not admit spectral singularities and cannot function as a laser unless the real part of their permittivity equals that of the vacuum. This restriction does not apply to doublets displaying anti-$\mathcal{P}\mathcal{T}$-symmetry. We determine the lasing threshold for a generic anti-$\mathcal{P}\mathcal{T}$-symmetric doublet and show that it possesses a continuous lasing spectrum.

Optical realization of one-dimensional generalized split-step quantum walks

Quantum walks are more than tools for building quantum algorithms. They have been used effectively to model and simulate quantum dynamics in many complex physical processes. Particularly, a variant of discrete-time quantum walk known as split-step quantum walk is closely related to Dirac cellular automata and topological insulators, whose realizations rely on position-dependent control of evolution operators. Owing to the ease of manipulating multiple degrees of freedom of photons, we provide an optical setup of split-step operators which, in combination with position-dependent coin (PDC) operation, can accomplish a table-top setup of generalized split-step walks. Also, we propose an optical implementation for PDC operation that allows, for instance, realizing electric quantum walks, control localization dynamics, and emulate space-time curvature effects. In addition, we propose a setup to realize any t-step split-step quantum walk involving 2 J-plates, 2 variable waveplates, a half-waveplate, an optical switch, and an optical delay line.

Invariant-Parameterized Exact Evolution Operator for SU(2) Systems with Time-Dependent Hamiltonian.

We report the step-by-step construction of the exact, closed and explicit expression for the evolution operator U(t) of a localized and isolated qubit in an arbitrary time-dependent field, which for concreteness we assume to be a magnetic field. Our approach is based on the existence of two independent dynamical invariants that enter the expression of SU(2) by means of two strictly related time-dependent, real or complex, parameters. The usefulness of our approach is demonstrated by exactly solving the quantum dynamics of a qubit subject to a controllable time-dependent field that can be realized in the laboratory. We further discuss possible applications to any SU(2) model, as well as the applicability of our method to realistic physical scenarios with different symmetry properties.

Quantum computing for fusion energy science applications

This is a review of recent research exploring and extending present-day quantum computing capabilities for fusion energy science applications. We begin with a brief tutorial on both ideal and open quantum dynamics, universal quantum computation, and quantum algorithms. Then, we explore the topic of using quantum computers to simulate both linear and nonlinear dynamics in greater detail. Because quantum computers can only efficiently perform linear operations on the quantum state, it is challenging to perform nonlinear operations that are generically required to describe the nonlinear differential equations of interest. In this work, we extend previous results on embedding nonlinear systems within linear systems by explicitly deriving the connection between the Koopman evolution operator, the Perron–Frobenius evolution operator, and the Koopman–von Neumann evolution (KvN) operator. We also explicitly derive the connection between the Koopman and Carleman approaches to embedding. Extension of the KvN framework to the complex-analytic setting relevant to Carleman embedding, and the proof that different choices of complex analytic reproducing kernel Hilbert spaces depend on the choice of Hilbert space metric are covered in the appendixes. Finally, we conclude with a review of recent quantum hardware implementations of algorithms on present-day quantum hardware platforms that may one day be accelerated through Hamiltonian simulation. We discuss the simulation of toy models of wave–particle interactions through the simulation of quantum maps and of wave–wave interactions important in nonlinear plasma dynamics.

Diffusive size-structured population model with time-varying diffusion rate

<p style='text-indent:20px;'>This work is devoted to the study of a size-structured population model with a time-varying diffusion rate. Due to the seasonal variation, it is natural to consider the time-varying diffusion rate. Moreover, introducing the time-varying diffusion rate makes the model more challenging and requires the results of evolution operators for the analysis. Under some assumptions on the time-dependent diffusion coefficient, the existence and uniqueness of mild solution is shown. We apply the method of characteristics and evolution operators to derive our results. Positivity and boundedness of mild solution is also shown. Some examples are provided to illustrate the theoretical findings. We also solve our model numerically to study the size and spatial dynamics of population density.</p>

Existence and uniqueness of global solutions for non-autonomous evolution equations with state-dependent nonlocal conditions

In this paper, we consider the existence and uniqueness of global solutions for non-autonomous evolution equations with state-dependent nonlocal conditions, in which the undelayed part admits an evolution operator. We discuss the problems by utilizing theory of evolution operators, Schauder fixed point theorem and Banach fixed point theorem. Some new results on existence and uniqueness of solutions of the considered equation are obtained on the infinite internal [0,+?). In the end, the obtained results are applied to a class of non-autonomous heat equations with state-dependent nonlocal conditions.

Quantum evolution speed induced by Hamiltonian

In the issue of quantum evolution, quantum evolution speed is usually quantified by the time rate of change of state distance between the initial sate and its time evolution. In this paper, the path distance of quantum evolution is introduced to study the evolution of a quantum system, through the approach combined with basic theory of quantum evolution and the linear algebra. In a quantum unitary system, the quantum evolution operator contains the path information of the quantum evolution, where the path distance is determined by the principal argument of the eigenvalues of the unitary operator. Accordingly, the instantaneous quantum evolution speed is proportional to the distance between the maximum and minimum eigenvalues of the Hamiltonian. As one of the applications, the path distance and the instantaneous quantum evolution speed could be used to form a new lower bound of the real evolution time, which depends on the evolution operator and Hamiltonian, and is independent of the initial state. It is found that the lower bound presented here is exactly equal to the real evolution time in the range <inline-formula><tex-math id="M2">\begin{document}$ \left[ {0, {\pi }/({{2{\omega _{\rm{H}}}}}}) \right]$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="22-20231009_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="22-20231009_M2.png"/></alternatives></inline-formula>. The tool of path distance and instantaneous quantum evolution speed introduced here provides new method for the related researches.