Hermitian Matrices Research Articles

Eigenvalues of Dual Hermitian Matrices with Application in Formation Control

Eigenvalues of Dual Hermitian Matrices with Application in Formation Control

A qubit-efficient variational selected configuration-interaction method

Abstract Finding the ground-state energy of molecules is an important and challenging computational problem for which quantum computing can potentially find efficient solutions. The variational quantum eigensolver (VQE) is a quantum algorithm that tackles the molecular groundstate problem and is regarded as one of the flagships of quantum computing. Yet, to date, only very small molecules were computed via VQE, due to high noise levels in current quantum devices. Here we present an alternative variational quantum scheme that requires significantly less qubits than VQE. The reduction in the qubit number allows for shallower circuits to be sufficient, rendering the method more resistant to noise. The proposed algorithm, termed variational quantum selected-configuration-interaction (VQ-SCI), is based on: (a) representing the target groundstate as a superposition of Slater determinant configurations, encoded directly upon the quantum computational basis states; and (b) selecting a-priory only the most dominant configurations. This is demonstrated through a set of groundstate calculations of the H2, LiH, BeH2, H2O, NH3 and C2H4 molecules in the sto-3g basis set, performed on IBM quantum devices. We show that the VQ-SCI reaches the full configuration interaction energy within chemical accuracy using the lowest number of qubits reported to date. Moreover, when the SCI matrix is generated ‘on the fly’, the VQ-SCI requires exponentially less memory than classical SCI methods. This offers a potential remedy to a severe memory bottleneck problem in classical SCI calculations. Finally, the proposed scheme is general and can be straightforwardly applied for finding the groundstate of any Hermitian matrix, outside the chemical context.

Computing the matrix exponential with the double exponential formula

Abstract This article considers the computation of the matrix exponential e A {{\rm{e}}}^{A} with numerical quadrature. Although several quadrature-based algorithms have been proposed, they focus on (near) Hermitian matrices. In order to deal with non-Hermitian matrices, we use another integral representation including an oscillatory term and consider applying the double exponential (DE) formula specialized to Fourier integrals. The DE formula transforms the given integral into another integral whose interval is infinite, and therefore, it is necessary to truncate the infinite interval. In this article, to utilize the DE formula, we analyze the truncation error and propose two algorithms. The first one approximates e A {{\rm{e}}}^{A} with the fixed mesh size, which is a parameter in the DE formula affecting the accuracy. The second one computes e A {{\rm{e}}}^{A} based on the first one with automatic selection of the mesh size depending on the given error tolerance.

Generalized matrix functions and Hermitian matrices

Generalized matrix functions and Hermitian matrices

Higher order constraints for the (β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-deformed) Hermitian matrix models

We construct the (β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-deformed) higher order total derivative operators and analyze their remarkable properties. In terms of these operators, we derive the higher order constraints for the (β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-deformed) Hermitian matrix models. We prove that these (β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-deformed) higher order constraints are reducible to the Virasoro constraints. Meanwhile, the Itoyama–Matsuo conjecture for the constraints of the Hermitian matrix model is proved. We also find that through rescaling variable transformations, two sets of the constraint operators become the W-operators of W-representations for the (β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-deformed) partition function hierarchies in the literature.

Exceptional Periodicity and Magic Star algebras: Generalized roots, gradings, Hermitian Vinberg T-algebras and their derivations

We introduce countably infinite series of finite dimensional generalizations of the exceptional Lie algebras: in fact, each exceptional Lie algebra (but g2) is the first element of an infinite series of finite dimensional algebras, which we name Magic Star algebras. All these algebras (but the first elements of the infinite series) are not Lie algebras, but nevertheless they have remarkable similarities with many characterizing features of the exceptional Lie algebras; they also enjoy a kind of periodicity (inherited by Bott periodicity), which we name Exceptional Periodicity. We analyze the graded algebraic structures arising in a certain projection (named Magic Star projection) of the generalized root systems pertaining to Magic Star algebras, and we highlight the occurrence of a class of rank-3, Hermitian matrix (special Vinberg T)-algebras (which we call H algebras) on each vertex of such a projection. We then focus on the Magic Star algebra f4(n), which generalizes the non-simply laced exceptional Lie algebra f4, and deserves a treatment apart. Finally, we compute the Lie algebra of the inner derivations of the H algebras, pointing out the enhancements occurring for each first element of the series of Magic Star algebras, thus retrieving the result known for the derivations of cubic simple Jordan algebras.

The exact solution of the Wegner flow equation with the Mielke generator for 3 × 3 hermitian matrices

The Wegner flow equation is a differential equation for a family of Hamiltonians and can be considered as a continuous unitary transformation. In essence, the transformation continuously decouples degrees of freedom and gradually converts the transformed Hamiltonian into the near-diagonal form. The Wegner flow equation has so far been applied to many areas of physics, but mainly in perturbative calculations. However, the exact solutions are known only for the simplest of two-dimensional case and this certainly limits the practical application of the method. On the other hand, the knowledge of exact solutions is essential for many physical problems as well as it can help in constructing improved approximated approaches. In this paper we present the exact solution of the Wegner flow equation with the Mielke generator for 3 × 3 Hermitian matrices, the general solutions for N × N tridiagonal Hermitian matrices and partially for 4 × 4 real symmetric matrices.

Some New Algebraic Method Developments in the Characterization of Matrix Equalities

Algebraic expressions and equalities can be constructed arbitrarily in a given algebraic framework according to the operational rules provided, and thus it is a prominent and necessary task in mathematics and applications to construct, classify, and characterize various simple general algebraic expressions and equalities. As an update to this prominent topic in matrix algebra, this article reviews and improves upon the well-known block matrix methodology and matrix rank methodology in the construction and characterization of matrix equalities. We present a collection of fundamental and useful formulas for calculating the ranks of a wide range of block matrices and then derive from these rank formulas various valuable consequences. In particular, we present several groups of equivalent conditions in the characterizations of the Hermitian matrix, the skew-Hermitian matrix, the normal matrix, etc.

Normalization of ZZ instanton amplitudes in type 0B minimal superstring theory

Abstract We study ZZ instanton corrections in the (2, 4k) $$ \mathcal{N} $$ N = 1 minimal superstring theory with the type 0B GSO projection, which becomes the type 0B $$ \mathcal{N} $$ N = 1 super-JT gravity in the k → ∞ limit. Each member of the (2, 4k) family of theories has two phases distinguished by the sign of the Liouville bulk cosmological constant. The worldsheet method for computing the one-loop normalization constant multiplying the instanton corrections gives an ill-defined answer in both phases. We fix these divergences using insights from string field theory and find finite, unambiguous results. Each member of the (2, 4k) family of theories is dual to a double-scaled one-matrix integral, where the double-scaling limit can be obtained starting either from a unitary matrix integral with a leading one-cut saddle point, or from a hermitian matrix integral with a leading two-cut saddle point. The matrix integral exhibits a gap-closing transition, which is the same as the double-scaled Gross-Witten-Wadia transition when k = 1. We also compute instanton corrections in the double-scaled matrix integral for all k and in both phases, and find perfect agreement with the string theory results.

Representation computation for the hypergeometric function of a Hermitian matrix argument

We establish the exact expressions for the hypergeometric function of a Hermitian matrix argument. This result allows for the eigenvalues of the matrix argument to occur with arbitrary multiplicities and can be used for numerical computation. These exact expressions are particularly important since they provide the key ingredient which allows many results which involve this function to be useful from a practical engineering perspective.

Information scrambling and chaos induced by a Hermitian matrix

Abstract Given an
arbitrary \(V \times V\) Hermitian matrix,
considered as a finite discrete quantum Hamiltonian,
we use methods from graph and
ergodic theories to
construct a \textit{quantum Poincar'e map} at energy \(E\) and a
corresponding stochastic
\textit{classical Poincar'e-Markov map} at the same energy
on an appropriate
discrete
\textit{phase space}.
This phase space
consists of the directed edges of a
graph with \(V\) vertices that are in one-to-one correspondence with the
non-vanishing off-diagonal elements of \(H\).
The correspondence between quantum Poincar'e map and classical
Poincar'e-Markov map is
an alternative to the standard quantum-classical
correspondence based on a classical limit \(\hbar \to 0\). Most
importantly it can be constructed where no such limit exists.
Using standard methods from ergodic theory
we then proceed to define
an expression for the
\textit{Lyapunov exponent} \(\Lambda(E)\) of the classical map.
It measures the rate of loss of classical information in the dynamics
and relates it to the separation of stochastic \textit{classical
trajectories} in the phase space. We suggest that
 loss of information in the underlying
classical dynamics is an indicator for quantum information scrambling.

Calculable neutrino Dirac mass matrix and one-loop θ¯ in the minimal left-right symmetric model

We revisit the contribution to the strong CP parameter θ¯ from leptonic CP violation at one-loop level in the minimal left-right symmetric model in the case of generalized parity as the left-right symmetry. The Hermitian neutrino Dirac mass matrix MD can be calculated using the light and heavy neutrino masses and mixings. We propose a parametrization of the right-handed neutrino mixing matrix VR and construct the heavy neutrino mass that maintains the Hermiticity of MD. We further apply it to evaluate the one-loop θ¯, denoted as θ¯loop, as a function of the sterile neutrino masses for explicit examples of VR. By requiring the magnitude of θ¯loop≲10−10, we derive the upper limits on the sterile neutrino masses, which are within reach of direct searches at the Large Hadron Collider and neutrinoless double beta decay experiments. Furthermore, our parametrization is pertinent for other phenomenological investigations. Published by the American Physical Society 2024

On a Matrix Trace Inequality for Hermitian Matrices

On a Matrix Trace Inequality for Hermitian Matrices

A square-root speedup for finding the smallest eigenvalue

We describe a quantum algorithm for finding the smallest eigenvalue of a Hermitian matrix. This algorithm combines quantum phase estimation and quantum amplitude estimation to achieve a quadratic speedup with respect to the best classical algorithm in terms of matrix dimensionality, i.e. O~(N/ε) 9 9In this work O~ ignores terms that are polylogarithmic in N or 1/ε . black-box queries to an oracle encoding the matrix, where N is the matrix dimension and ɛ is the desired precision. In contrast, the best classical algorithm for the same task requires Ω(N)polylog(1/ε) queries. In addition, this algorithm allows the user to select any constant success probability. We also provide a similar algorithm with the same runtime that allows us to prepare a quantum state lying mostly in the matrix’s low-energy subspace. We implement simulations of both algorithms and demonstrate their application to problems in quantum chemistry and materials science.

Wavelet Collocation Method for HIV-1/HTLV-I Co-Infection Model Using Hermite Polynomial.

In this study, the dynamic behavior of fractional order co-infection model with human immunodeficiency virus type 1 (HIV-1) and human T-lymphotropic virus type I (HTLV-I) is analyzed using operational matrix of Hermite wavelet collocation method. Also, the uniqueness and existence of solutions are calculated based on the fixed point hypothesis. For the fractional order co-infection model, its positivity and boundedness are demonstrated. Furthermore, different types of Ulam-Hyres stability are also discussed. The numerical solution of the model are obtained by using the operational matrix of the Hermite wavelet approach. This scheme is used to solve the system of nonlinear equationsthat are very fruitful and easy to implement. Additionally, the stability analysis of the numerical scheme is explained. The mathematical model taken in this work incorporates the biological characteristics of both HIV-1 and HTLV-I. After that all the equilibrium points of the fractional order co-infection model are found and their existence conditions are explored with the help of the Caputo derivative. The global stability of all equilibrium points of this model are determined with the help of Lyapunov functions and the LaSalle invariance principle. Convergence analysis is also discussed. Hermite wavelet operational matrix methods are more accurate and convergent than other numerical methods. Lastly, variations in model dynamics are found when examining different fractional order values. These findings will be valuable to biologists in the treatment of HIV-1/HTLV-I.

Random anti-commuting Hermitian matrices

We consider pairs of anti-commuting [Formula: see text]-by-[Formula: see text] Hermitian matrices that are chosen randomly with respect to a Gaussian measure. Generically such a pair decomposes into the direct sum of [Formula: see text]-by-[Formula: see text] blocks on which the first matrix has eigenvalues [Formula: see text] and the second has eigenvalues [Formula: see text]. We call [Formula: see text] the skew spectrum of the pair. We derive a formula for the probability density of the skew spectrum, and show that the elements are repelling.

On the signs of the principal minors of Hermitian matrices

The signed enhanced principal rank characteristic sequence (sepr-sequence) of a given n × n Hermitian matrix B is the sequence t 1 t 2 … t n , where t k is A ∗ , A + , A − , N , S ∗ , S + , or S − , based on the following criteria: t k = A ∗ if all the order-k principal minors of B are nonzero, and two of those minors are of opposite sign; t k = A + (respectively, t k = A − ) if all the order-k principal minors of B are positive (respectively, negative); t k = N if all the order-k principal minors of B are zero; t k = S ∗ if B has a positive, a negative, and a zero order-k principal minor; t k = S + (respectively, t k = S − ) if B has both a zero and a nonzero order-k principal minor, and all the nonzero order-k principal minors of B are positive (respectively, negative). A complete characterization of the sequences of order 2 and order 3 that do not occur as a subsequence of the sepr-sequence of any Hermitian matrix is presented (a sequence has order k if it has k terms). An analogous characterization for real symmetric matrices is presented as well.

Functional renormalization group for “p = 2” like glassy matrices in the planar approximation II. Ward identities method in the deep IR

This paper, as a continuation of our previous investigation [Nucl. Phys. B 1005 (2024) 116582] aims to study the glassy random matrices with quenched Wigner disorder. In this previous work, we have constructed a renormalization group based on the effective deterministic kinetic spectrum emerging from large N limit, and we extended approximate solutions using standard vertex expansion, at the leading order of the derivative expansion. Now in the following work, by introducing the non-trivial Ward identities which come from the (U(N))×2 symmetry broken of the effective kinetic action, we provide in the deep IR the explicit solution of the functional renormalization group for a model with quartic coupling by solving the Hierarchy to all orders in the local sector, which in particular imply the vanishing of the anomalous dimension. The numerical investigations confirm the first-order phase transition discovered in the vertex expansion framework, both in the active and passive schemes. Finally, we extend the discussion to hermitian matrices.

Fundamental Limits in Measuring the Anisotropic Rotational Diffusion of Single Molecules.

Many biophysical techniques, such as single-molecule fluorescence correlation spectroscopy, Förster resonance energy transfer, and fluorescence anisotropy, measure the translation and rotation of biomolecules to quantify molecular processes at the nanoscale. These methods often simplify data analysis by assuming isotropic rotational diffusion, e.g., that molecules wobble within a circular cone. This simplification ignores the anisotropy present in many biological contexts that may cause molecules to exhibit different degrees of diffusion in different directions. Here, we loosen this assumption and establish a theoretical framework for describing and measuring anisotropic rotational diffusion using fluorescence imaging. We show that anisotropic wobble is directly quantified by the eigenvalues of a 3-by-3 positive-semidefinite Hermitian matrix M consisting of the second-order moments of a molecule's transition dipole μ. This formalism enables us to model the influence of unavoidable shot noise using a Hermitian perturbation matrix E; the eigenvalues of E directly bound errors in measurements of wobble via Weyl's inequality. Quantifying various perturbations E reveals that anisotropic wobble measurements are generally more sensitive to errors compared to quantifying isotropic wobble. Moreover, severe shot noise can induce negative eigenvalues in estimates of M, thereby causing the anisotropic wobble measurement to fail. Our analysis, using Fisher information, shows that techniques with worse orientation measurement sensitivity experience stronger perturbations E and require larger signal to background ratios to measure anisotropic rotational diffusion accurately. Our work provides deep insights for improving the state of the art in imaging the orientations and anisotropic rotational diffusion of single molecules.

Level statistics of the one-dimensional dimerized Hubbard model

The statistical properties of level spacings provide valuable insights into the dynamical properties of a many-body quantum systems. We investigate the level statistics of the Fermi–Hubbard model with dimerized hopping amplitude and find that after taking into account translation, reflection, spin and η pairing symmetries to isolate irreducible blocks of the Hamiltonian, the level spacings in the limit of large system sizes follow the distribution expected for hermitian random matrices from the Gaussian orthogonal ensemble. We show this by analyzing the distribution of the ratios of consecutive level spacings in this system, its cumulative distribution and quantify the deviations of the distributions using their mean, standard deviation and skewness.