Nonorthogonal Basis Research Articles

R -matrix theory with level-dependent boundary condition parameters

I present a new formalism of the R-matrix theory where the formal parameters for the resonance energies and widths are identical to the observed values. By allowing the boundary condition parameters to vary from level to level, the freedom required to adjust the formal parameters for the pole positions to the observed values is obtained. The basis of the resulting theory becomes nonorthogonal, and I describe the procedure to construct a consistent R-matrix theory with such a nonorthogonal basis. And by adjusting the normalization of the states that form the basis, the formal parameters for the reduced decay widths also become the same as those observed, leaving no formal parameters that are different from the observed ones. A demonstration of the developed theory to the elastic 12C+p scattering data is presented.

A reference tissue forward model for improved PET accuracy using within-scan displacement studies

We report a novel forward-model implementation of the full reference tissue model (fFTRM) that addresses the fast-exchange approximation employed by the simplified reference tissue model (SRTM) by incorporating a non-zero dissociation time constant from the specifically bound compartment. The forward computational approach avoided errors associated with noisy and nonorthogonal basis functions using an inverse linear model. Compared to analysis by a multilinear single-compartment reference tissue model (MRTM), fFTRM provided improved accuracy for estimation of binding potentials at early times in the scan, with no worse reproducibility across sessions. To test the model’s ability to identify small focal changes in binding potential using a within-scan challenge, we employed a nonhuman primate model of focal dopamine release elicited by deep brain microstimulation remote to ventral striatum (VST) during imaging by simultaneous PET and fMRI. The new model reported an unambiguously lateralized response in VST consistent with fMRI, whereas the MRTM-derived response was not lateralized and was consistent with simulations of model bias. The proposed model enabled better accuracy in PET [11C]raclopride displacement studies and may also facilitate challenges sooner after injection, thereby recovering some sensitivity lost to radioactive decay of the PET tracer.

Simplified State Interaction for Matrix Product State Wave Functions.

We present an approximation to the state-interaction approach for matrix product state (MPS) wave functions (MPSSI) in a nonorthogonal molecular orbital basis, first presented by Knecht et al. [J. Chem. Theory Comput.,2016, 28, 5881], that allows for a significant reduction of the computational cost without significantly compromising its accuracy. The approximation is well-suited if the molecular orbital basis is close to orthogonality, and its reliability may be estimated a priori with a single numerical parameter. For an example of a platinum azide complex, our approximation offers up to 63-fold reduction in computational time compared to the original method for wave function overlaps and spin–orbit couplings, while still maintaining numerical accuracy.

Even More Efficient Quantum Computations of Chemistry Through Tensor Hypercontraction

We describe quantum circuits with only O~(N) Toffoli complexity that block encode the spectra of quantum chemistry Hamiltonians in a basis of N arbitrary (e.g., molecular) orbitals. With O(λ/ϵ) repetitions of these circuits one can use phase estimation to sample in the molecular eigenbasis, where λ is the 1-norm of Hamiltonian coefficients and ϵ is the target precision. This is the lowest complexity shown for quantum computations of chemistry within an arbitrary basis. Furthermore, up to logarithmic factors, this matches the scaling of the most efficient prior block encodings that can work only with orthogonal-basis functions diagonalizing the Coloumb operator (e.g., the plane-wave dual basis). Our key insight is to factorize the Hamiltonian using a method known as tensor hypercontraction (THC) and then to transform the Coulomb operator into an isospectral diagonal form with a nonorthogonal basis defined by the THC factors. We then use qubitization to simulate the nonorthogonal THC Hamiltonian, in a fashion that avoids most complications of the nonorthogonal basis. We also reanalyze and reduce the cost of several of the best prior algorithms for these simulations in order to facilitate a clear comparison to the present work. In addition to having lower asymptotic scaling space-time volume, compilation of our algorithm for challenging finite-sized molecules such as FeMoCo reveals that our method requires the least fault-tolerant resources of any known approach. By laying out and optimizing the surface-code resources required of our approach we show that FeMoCo can be simulated using about four million physical qubits and under 4 days of runtime, assuming 1-μs cycle times and physical gate-error rates no worse than 0.1%.13 MoreReceived 12 December 2020Revised 7 April 2021Accepted 24 May 2021DOI:https://doi.org/10.1103/PRXQuantum.2.030305Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum algorithmsQuantum computationQuantum simulationPhysical SystemsStrongly correlated systemsTechniquesElectronic structureCondensed Matter, Materials & Applied PhysicsQuantum InformationAtomic, Molecular & Optical

Adaptive Beamforming for Passive Synthetic Aperture with Uncertain Curvilinear Trajectory

Recently, numerous reconstruction-based adaptive beamformers have been proposed, which can improve the quality of imaging or localization in the application of passive synthetic aperture (PSA) sensing. However, when the trajectory is curvilinear, existing beamformers may not be robust enough to suppress interferences efficiently. To overcome the model mismatch of unknown curvilinear trajectory, this paper presents an adaptive beamforming algorithm by reconstructing the interference-plus-noise covariance matrix (INCM). Using the idea of signal subspace fitting, we construct a joint optimization problem, where the unknown directions of arrival (DOAs) and array shape parameters are coupled together. To tackle this problem, we develop a hybrid optimization method by combining the genetic algorithm and difference-based quasi-Newton method. Then, a set of non-orthogonal bases for signal subspace is estimated with an acceptable computational complexity. Instead of reconstructing the covariance matrix by integrating the spatial spectrum over interference angular sector, we extract the desired signal covariance matrix (DSCM) directly from signal subspace, and then the INCM is reconstructed by eliminating DSCM from the sample covariance matrix (SCM). Numerical simulations demonstrate the robustness of the proposed beamformer in the case of signal direction error, local scattering and random curvilinear trajectory.

Learning Set Functions that are Sparse in Non-Orthogonal Fourier Bases

Many applications of machine learning on discrete domains, such as learning preference functions in recommender systems or auctions, can be reduced to estimating a set function that is sparse in the Fourier domain. In this work, we present a new family of algorithms for learning Fourier-sparse set functions. They require at most nk − k log k + k queries (set function evaluations), under mild conditions on the Fourier coefficients, where n is the size of the ground set and k the number of non-zero Fourier coefficients. In contrast to other work that focused on the orthogonal Walsh-Hadamard transform (WHT), our novel algorithms operate with recently introduced non-orthogonal Fourier transforms that offer different notions of Fourier-sparsity. These naturally arise when modeling, e.g., sets of items forming substitutes and complements. We demonstrate effectiveness on several real-world applications.

Low-rank matrix completion in a general non-orthogonal basis

This paper considers theoretical analysis of recovering a low rank matrix given a few expansion coefficients with respect to any basis. The current approach generalizes the existing analysis for the low-rank matrix completion problem with sampling under entry sensing or with respect to a symmetric orthonormal basis. The analysis is based on dual certificates using a dual basis approach. We introduce a condition on the basis called the correlation condition. This condition can be computed in time O(n3) and holds for many cases of deterministic basis. If the correlation condition holds and the underlying low rank matrix obeys the coherence condition with parameter ν, under additional mild assumptions, our main result shows that the true matrix can be recovered with very high probability from O(nrνlog2⁡n) uniformly random expansion coefficients.

Central moment lattice Boltzmann method on a rectangular lattice

Simulating inhomogeneous flows with different characteristic scales in different coordinate directions using the collide-and-stream based lattice Boltzmann methods (LBMs) can be accomplished efficiently using rectangular lattice grids. We develop and investigate a new rectangular central moment LBM based on a non-orthogonal moment basis and involving multiple relaxation times. The equilibria to which the central moments relax under collision in this approach are obtained from matching with those corresponding to the continuous Maxwell distribution. A Chapman–Enskog analysis is performed to derive the correction terms to the second order moment equilibria involving the grid aspect ratio and velocity gradients that restore the isotropy of the viscous stress tensor and eliminate the non-Galilean invariant cubic velocity terms in the resulting hydrodynamical equations. A special case of this rectangular formulation involving the raw moments is also constructed. The resulting schemes represent a considerable simplification, especially for the transformation matrices and isotropy corrections, and improvement over the existing lattice Boltzmann schemes based on raw moments on rectangular lattice grids that use orthogonal moment basis. Numerical validation study of both the proposed rectangular LBMs for a variety of benchmark flows is performed, which shows good accuracy at various grid aspect ratios. The ability of our proposed schemes to simulate flows at relatively low grid aspect ratios and higher Reynolds numbers than considered in prior approaches is demonstrated. Furthermore, simulations reveal the superior stability characteristics of the rectangular central moment LBM over that based on raw moments in handling shear flows at lower viscosities and/or higher characteristic velocities. In addition, computational advantages of using our rectangular LB formulation in lieu of that based on the square lattice are shown.

New and Practical Formulation for Overlaps of Bogoliubov Vacua.

In this Letter, we present a new expression for the overlaps of wave functions in Hartree-Fock-Bogoliubov based theories. Starting from the Pfaffian formula by Bertsch etal. [1], an exact and computationally stable formula for overlaps is derived. We illustrate the convenience of this new formulation with a numerical application in the context of the particle-number projection method. This new formula allows for substantially increased precision and versatility in chemical, atomic, and nuclear physics applications, particularly for methods dealing with superfluidity, symmetry restoration, and uses of nonorthogonal many-body basis states.

The Use of Contravariant Tensors to Model Anisotropic Soft Tissues

Biological tissues have been shown to behave isotropically at lower strain values, while at higher strains the fibers embedded in the tissue straighten and tend to take up the load. Thus, the anisotropy induced at higher loads can be mathematically modeled by incorporating the strains experienced by the fibers. From histological studies on soft tissues it is evident that for a wide range of tissues the fibers have an oblique mean orientation about the physiological loading directions. Thus, we require a mathematical framework of tensors defined in nonorthogonal basis to capture the direction-dependent response of fibers under high induced loads. In this work, we propose a novel approach to determine the fiber strains with the aid of the contravariant tensors defined in an oblique coordinate system. To determine the fiber strains, we introduce a fourth-order contravariant fiber orientation transformation tensor. The approach helps us successfully in determining the fiber strains, for a family of symmetrically and asymmetrically oriented fibers, with the aid of a single anisotropic invariant. The proposed model was fitted with the experimental data from literature to determine the corresponding material parameters.

Machine learning in spectral domain

Deep neural networks are usually trained in the space of the nodes, by adjusting the weights of existing links via suitable optimization protocols. We here propose a radically new approach which anchors the learning process to reciprocal space. Specifically, the training acts on the spectral domain and seeks to modify the eigenvalues and eigenvectors of transfer operators in direct space. The proposed method is ductile and can be tailored to return either linear or non-linear classifiers. Adjusting the eigenvalues, when freezing the eigenvectors entries, yields performances that are superior to those attained with standard methods restricted to operate with an identical number of free parameters. To recover a feed-forward architecture in direct space, we have postulated a nested indentation of the eigenvectors. Different non-orthogonal basis could be employed to export the spectral learning to other frameworks, as e.g. reservoir computing.

Experimental demonstrations of unconditional security in a purely classical regime

So far, unconditional security in key distribution processes has been confined to quantum key distribution (QKD) protocols based on the no-cloning theorem of nonorthogonal bases. Recently, a completely different approach, the unconditionally secured classical key distribution (USCKD), has been proposed for unconditional security in the purely classical regime. Unlike QKD, both classical channels and orthogonal bases are key ingredients in USCKD, where unconditional security is provided by deterministic randomness via path superposition-based reversible unitary transformations in a coupled Mach–Zehnder interferometer. Here, the first experimental demonstration of the USCKD protocol is presented.

The Sternheimer approach to all-electron real-space density-functional perturbation theory with atomic basis set

We present an efficient perturbative method to get the response density matrix using localized non-orthogonal basis sets. This scheme is based on the solution of the coupled perturbed self-consistent field equation with the Sternheimer approach, which only requires the occupied states and avoids a sum over unoccupied states. We present a complete derivation of the Sternheimer approach to perturbation theory within the framework of the linear combination of atomic orbitals. To demonstrate the capabilities of this method, we have implemented it in the all-electron Fritz Haber Institute ab initio molecular simulation package and applied it to benchmark molecules. For the response properties with respect to the atomic displacement and to the homogeneous electric field, the results are in excellent agreement with those of the previous traditional method and fully validate this Sternheimer approach.

BB84 with several intercepts and resend attacks using partially non-orthogonal basis states through a depolarising channel

The paper presents the partially non-orthogonal basis states effect in presence of several intercept and resend attacks through a depolarising channel. The quantum error and the mutual information are computed for arbitrary angles of the non-orthogonal basis states and the depolarising channel parameter. It is found that the information security depends strongly on the non-orthogonal basis states angle, the depolarising channel parameter, the intercepts and resend attack probability and the eavesdropper number. Besides, it is found that for any eavesdropper's number and basis states angle, the protocol is more secure if the depolarisation parameter is lower than a threshold value and the maximum value of the quantum error decreases by increasing the depolarising parameter and/or the eavesdropper's number. While for the depolarisation parameter above the threshold value the protocol is not secured independently of any values of the non-orthogonal basis states angle and/or the eavesdroppers' number.

BB84 with several intercepts and resend attacks using partially non-orthogonal basis states through a depolarising channel

BB84 with several intercepts and resend attacks using partially non-orthogonal basis states through a depolarising channel

A general higher‐order shell theory for compressible isotropic hyperelastic materials using orthonormal moving frame

SummaryThe primary objective of this study is threefold: (1) to present a general higher‐order shell theory to analyze large deformations of thin or thick shell structures made of general compressible hyperelastic materials; (2) to formulate an efficient shell theory using the orthonormal moving frame, and (3) to develop and apply the nonlinear weak‐form Galerkin finite element model for the proposed shell theory. The displacement field of the line normal to the shell reference surface is approximated by the Taylor series/Legendre polynomials in the thickness coordinate of the shell. The use of an orthonormal moving frame makes it possible to represent kinematic quantities (e.g., the determinant of the deformation gradient) in a far more efficient manner compared with the nonorthogonal covariant bases. Kinematic quantities for the shell deformation are obtained in a novel way in the surface coordinate described in the appendix of this study with the help of exterior calculus. Furthermore, the governing equation of the shell deformation has been derived in the general surface coordinates. To obtain the nonlinear solution in the quasi‐static cases, we develop the weak‐form finite element model in which the reference surface of the shell is modeled exactly. The general invariant based compressible hyperelastic material model is considered. The formulation presented herein can be specialized for various other nonlinear compressible hyperelastic constitutive models, for example, in biomechanics and other soft‐material problems (e.g., compressible neo‐Hookean material, compressible Mooney–Rivlin material, Saint Venant–Kirchhoff model, and others). A number of numerical examples are presented to verify and validate the formulation presented in this study. The scope of potential extensions are outlined in the final section of this study.

Spectrum Shaping to Avoid OFDM Interference to In-Band Radars in the U-NII Bands

Interference avoidance will be an essential feature of wireless communication systems in the future. Interference avoidance to narrowband users is particularly difficult with Orthogonal Frequency Division Multiplexing (OFDM) due to the side lobes of the subcarriers. Spectrum-nulling algorithms based on cancellation signals have been studied in the literature. In this letter, we describe a general technique to design spectrum nulls for arbitrary signal and canceler bases. Appropriately chosen nonorthogonal canceler bases can result in data rate- and energy-efficient design of spectrum nulls. We also describe an iterative algorithm to cancel the resulting inter-carrier interference at the receiver. Spectrum nulls that meet requirements for avoiding interference to weather radars in the 5 GHz band can be designed using this technique. Bit error rates have been evaluated using simulations.

Theoretical evidence of a significant modification of the electronic structure of double-walled carbon nanotubes due to the interlayer interaction

The recently reported experimental optical spectra of double-walled carbon nanotubes exhibit more peaks than it could be expected based on the layers alone. The appearance of excess peaks has been attributed to the interlayer interaction. In order to elucidate the origin of the excess peaks, we perform calculations of the optical absorption of a particular nanotube using the recursion method with non-orthogonal tight-binding basis functions. Our study shows that the interlayer interaction can give rise to major changes in the electronic structure of this nanotube, manifesting themselves with shifts of the optical transitions and appearance of new optical transitions. The derived absorption spectrum is found to be in excellent agreement with the available experimental data, which justifies the use of the proposed approach for double-walled carbon nanotubes.

An efficient circle-type multiparty quantum key agreement protocol with single particles

An efficient circle-type multiparty quantum key agreement protocol utilizing two non-orthogonal bases single particles is proposed. In our protocol, each participant performs one of two simple unitary operations to encode and additional unitary operations to protect his privacy. Both outside attacks and participant attacks can be perfectly resisted. Moreover, our protocol has a high efficiency up to [Formula: see text], where [Formula: see text] is the number of the participants, and it can be easily implemented with current quantum technologies as it needs only single particle measurements.

On Data-Driven Sparse Sensing and Linear Estimation of Fluid Flows.

The reconstruction of fine-scale information from sparse data measured at irregular locations is often needed in many diverse applications, including numerous instances of practical fluid dynamics observed in natural environments. This need is driven by tasks such as data assimilation or the recovery of fine-scale knowledge including models from limited data. Sparse reconstruction is inherently badly represented when formulated as a linear estimation problem. Therefore, the most successful linear estimation approaches are better represented by recovering the full state on an encoded low-dimensional basis that effectively spans the data. Commonly used low-dimensional spaces include those characterized by orthogonal Fourier and data-driven proper orthogonal decomposition (POD) modes. This article deals with the use of linear estimation methods when one encounters a non-orthogonal basis. As a representative thought example, we focus on linear estimation using a basis from shallow extreme learning machine (ELM) autoencoder networks that are easy to learn but non-orthogonal and which certainly do not parsimoniously represent the data, thus requiring numerous sensors for effective reconstruction. In this paper, we present an efficient and robust framework for sparse data-driven sensor placement and the consequent recovery of the higher-resolution field of basis vectors. The performance improvements are illustrated through examples of fluid flows with varying complexity and benchmarked against well-known POD-based sparse recovery methods.