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This article presents a novel and succinct algorithmic framework via alternating quantum walks, unifying quantum spatial search, state transfer and uniform sampling on a large class of graphs. Using the framework, we can achieve exact uniform sampling over all vertices and perfect state transfer between any two vertices, provided that eigenvalues of Laplacian matrix of the graph are all integers. Furthermore, if the graph is vertex-transitive as well, then we can achieve deterministic quantum spatial search that finds a marked vertex with certainty. In contrast, existing quantum search algorithms generally has a certain probability of failure. Even if the graph is not vertex-transitive, such as the complete bipartite graph, we can still adjust the algorithmic framework to obtain deterministic spatial search, which thus shows the flexibility of it. Besides unifying and improving plenty of previous results, our work provides new results on more graphs. The approach is easy to use since it has a succinct formalism that depends only on the depth of the Laplacian eigenvalue set of the graph, and may shed light on the solution of more problems related to graphs., Comment: This manuscript has some overlap with arXiv:2307.16133. More precisely, it is an advanced version of arXiv:2307.16133, which not only modifies the paper structure and some results but also adds several new results

In this paper, we extend the Chen and Moore determinants of quaternion Hermitian} matrices to dual quaternion Hermitian matrices. We show the Chen determinant of dual quaternion Hermitian {matrices is invariant under addition, switching, multiplication, and unitary operations at the both hand sides. We then show the Chen and Moore determinants of dual quaternion Hermitian matrices are equal to each other, and they are also equal to the products of eigenvalues. The characteristic polynomial of a dual quaternion Hermitian matrix is also studied.

In the field of robotics research, a crucial applied problem is the hand-eye calibration issue. The corresponding matrix equation $AX=YB$ in hand-eye calibration can be equivalently transformed into the dual quaternion equation $q_Aq_X=q_Yq_B$. However, the dual quaternion equation $q_Aq_X=q_Yq_B$ is merely a specific case of the more general dual quaternion matrix equation $AX-YB=C$, which also holds significant applications in system and control theory. Therefore, we in this paper establish the solvability conditions of the dual quaternion matrix equation $AX-YB=C$ and provide a general expression for its solutions when it is solvable. As an application, we investigate the analytical solutions of the dual quaternion matrix equation $AX=YB$. Finally, we validate the main results of this paper through a numerical example.

Spatial search is an important problem in quantum computation, which aims to find a marked vertex on a graph. We propose a novel approach for designing deterministic quantum search algorithms on a variety of graphs via alternating quantum walks. Our approach is universal because it does not require an instance-specific analysis for different graphs. We highlight the flexibility of our approach by proving that for Johnson graphs, rook graphs, complete-square graphs and complete bipartite graphs, our quantum algorithms can find the marked vertex with $100\%$ success probability and achieve quadratic speedups over classical algorithms. This not only gives an alternative succinct way to prove the existing results, but also leads to new interesting findings on more general graphs., Comment: The language has been polished and the organization of the paper has been adjusted

This paper is aimed to identify some new characterizations and representations of the Minkowski inverse in Minkowski space. First of all, a few representations of {1,3m}, {1,2,3m}, {1,4m} and {1,2,4m}-inverses are given in order to represent the Minkowski inverse. Secondly, some famous characterizations of the Moore-Penrose inverse are extended to that of the Minkowski inverse. Thirdly, using the Hartwig-Spindelb\"ock decomposition we present a representation of the Minkowski inverse. And, based on this result, an interesting characterization of the Minkowski inverse is showed by a rank equation. Finally, we obtain several new representations of the Minkowski inverse in a more general form, by which the Minkowski inverse of a class of block matrices is given.

This paper first introduces a new generalized inverse in Minkowski space, called the m-DMP inverse, and discusses its algebraic and geometrical properties. The second objective is to characterize the m-DMP inverse equivalently by ranges, null spaces and matrix equations, and show its integral and limiting representations and several explicit expressions. Finally, the paper gives applications of the m-DMP inverse in solving a system of linear equations and a constrained optimization problem.

Many modern spin-based devices rely on the spin-orbit interaction, which is highly sensitive to the host semiconductor heterostructure and varies substantially depending on crystal direction, crystal asymmetry (Dresselhaus), and quantum confinement asymmetry (Rashba). One-dimensional quantum point contacts are a powerful tool to probe both energy and directional dependence of spin-orbit interaction through the effect on the hole $g$-factor. In this work we investigate the role of cubic crystal asymmetry in driving an oscillation in the in-plane hole $g$-factor anisotropy when the quantum point contact is rotated with respect to the crystal axes, and we are able to separate contributions to the Zeeman Hamiltonian arising from Rashba and cubic crystal asymmetry spin-orbit interactions. The in-plane $g$-factor is found to be extremely sensitive to the orientation of the quantum point contact, changing by a factor of $5$ when rotated by $45^{\circ}$. This exceptionally strong crystal lattice anisotropy of the in-plane Zeeman splitting cannot be explained within axially symmetric theoretical models. Theoretical modelling based on the combined Luttinger, Rashba and Dresselhaus Hamiltonians that we use here reveals new spin-orbit contributions to the in-plane hole $g$-factor and provides an excellent agreement with our experimental data., Comment: 8 pages, 4 figures

We propose a new approach toward reconstructing the late-time near-horizon geometry of merging binary black holes, and toward computing gravitational-wave echoes from exotic compact objects. A binary black-hole merger spacetime can be divided by a time-like hypersurface into a Black-Hole Perturbation (BHP) region, in which the space-time geometry can be approximated by homogeneous linear perturbations of the final Kerr black hole, and a nonlinear region. At late times, the boundary between the two regions is an infalling shell. The BHP region contains late-time gravitational-waves emitted toward the future horizon, as well as those emitted toward future null infinity. In this region, by imposing no-ingoing wave conditions at past null infinity, and matching out-going waves at future null infinity with waveforms computed from numerical relativity, we can obtain waves that travel toward the future horizon. In particular, the Newman-Penrose $\psi_0$ associated with the in-going wave on the horizon is related to tidal deformations measured by fiducial observers floating above the horizon. We further determine the boundary of the BHP region on the future horizon by imposing that $\psi_0$ inside the BHP region can be faithfully represented by quasi-normal modes. Using a physically-motivated way to impose boundary conditions near the horizon, and applying the so-called Boltzmann reflectivity, we compute the quasi-normal modes of non-rotating ECOs, as well as gravitational-wave echoes. We also investigate the detectability of these echoes in current and future detectors, and prospects for parameter estimation.

We present numerical waveforms of gravitational-wave echoes from spinning exotic compact objects (ECOs) that result from binary black hole coalescence. We obtain these echoes by solving the Teukolsky equation for the $\psi_4$ associated with gravitational waves that propagate toward the horizon of a Kerr spacetime, and process the subsequent reflections of the horizon-going wave by the surface of the ECO, which lies right above the Kerr horizon. The trajectories of the infalling objects are modified from Kerr geodesics, such that the gravitational waves propagating toward future null infinity match those from merging black holes with comparable masses. In this way, the corresponding echoes approximate to those from comparable-mass mergers. For boundary conditions at the ECO surface, we adopt recent work using the membrane paradigm, which relates $\psi_0$ associated with the horizon-going wave and $\psi_4$ of the wave that leaves the ECO surface. We obtain $\psi_0$ of the horizon-going wave from $\psi_4$ using the Teukolsky-Starobinsky relation. The echoes we obtain turn out to be significantly weaker than those from previous studies that generate echo waveforms by modeling the ringdown part of binary black hole coalescence waveforms as originating from the past horizon.

Teukolsky equations for $|s|=2$ provide efficient ways to solve for curvature perturbations around Kerr black holes. Imposing regularity conditions on these perturbations on the future (past) horizon corresponds to imposing an in-going (out-going) wave boundary condition. For exotic compact objects (ECOs) with external Kerr spacetime, however, it is not yet clear how to physically impose boundary conditions for curvature perturbations on their boundaries. We address this problem using the Membrane Paradigm, by considering a family of fiducial observers (FIDOs) that float right above the horizon of a linearly perturbed Kerr black hole. From the reference frame of these observers, the ECO will experience tidal perturbations due to in-going gravitational waves, respond to these waves, and generate out-going waves. As it also turns out, if both in-going and out-going waves exist near the horizon, the Newman Penrose (NP) quantity $\psi_0$ will be numerically dominated by the in-going wave, while the NP quantity $\psi_4$ will be dominated by the out-going wave. In this way, we obtain the ECO boundary condition in the form of a relation between $\psi_0$ and the complex conjugate of $\psi_4$, in a way that is determined by the ECO's tidal response in the FIDO frame. We explore several ways to modify gravitational-wave dispersion in the FIDO frame, and deduce the corresponding ECO boundary condition for Teukolsky functions. We subsequently obtain the boundary condition for $\psi_4$ alone, as well as for the Sasaki-Nakamura and Detweiler's functions. As it also turns out, reflection of spinning ECOs will generically mix between different $\ell$ components of the perturbations fields, and be different for perturbations with different parities. We also apply our boundary condition to computing gravitational-wave echoes from spinning ECOs, and solve for the spinning ECOs' quasi-normal modes., Comment: 20 pages, 6 figures

Optimization models involving quaternion matrices are widely used in color image process and other engineering areas. These models optimize real functions of quaternion matrix variables. In particular, $\ell_0$-norms and rank functions of quaternion matrices are discrete. Yet calculus with derivatives, subdifferentials and generalized subdifferentials of such real functions is needed to handle such models. In this paper, we introduce first and second order derivatives and establish their calculation rules for such real functions. Our approach is consistent with the subgradient concept for norms of quaternion matrix variables, recently introduced in the literature. We develop the concepts of generalized subdifferentials of proper functions of quaternion matrices, and use them to analyze the optimality conditions of a sparse low rank color image denoising model. We introduce R-product for two quaternion matrix vectors, as a key tool for our calculus. We show that the real representation set of low-rank quaternion matrices is closed and semi-algebraic. We also establish first order and second order optimality conditions for constrained optimization problems of real functions in quaternion matrix variables.

Black Holes are possibly the most enigmatic objects in our Universe. From their detection in gravitational waves upon their mergers, to their snapshot eating at the centres of galaxies, black hole astrophysics has undergone an observational renaissance in the past 4 years. Nevertheless, they remain active playgrounds for strong gravity and quantum effects, where novel aspects of the elusive theory of quantum gravity may be hard at work. In this review article, we provide an overview of the strong motivations for why "Quantum Black Holes" may be radically different from their classical counterparts in Einstein's General Relativity. We then discuss the observational signatures of quantum black holes, focusing on gravitational wave echoes as smoking guns for quantum horizons (or exotic compact objects), which have led to significant recent excitement and activity. We review the theoretical underpinning of gravitational wave echoes and critically examine the seemingly contradictory observational claims regarding their (non-)existence. Finally, we discuss the future theoretical and observational landscape for unraveling the "Quantum Black Holes in the Sky"., Comment: 78 pages, 42 figures, Invited review article for the special issue of the journal "Universe" on "Probing new physics with black holes", Comment are very welcome!