Real Matrix Elements Research Articles

Strategies for Variational Quantum Compiling of a Zero-Phase Beam Splitter on the Xanadu X8 Processor

In the context of variational compiling of a continuous-variable (CV) unitary operation, the architecture and parameter space of the Xanadu X8 processor constrain both the set of feasible compiling problems and the allowed cost functions. In this paper, we motivate a faithful cost function for variational compiling of a two-mode, continuous-variable beam splitter gate with real matrix elements (i.e., a “zero-phase” beam splitter) that complies with the constraints of the X8 processor. This cost function is then computed on the X8. Despite the noise in the processor, we find that the cost function exhibits optimum parameter resilience and, therefore, that this variational compiling problem is feasible on the X8. The intent of the paper is partly to report a proof-of-principle cost function calculation on near-term CV hardware and partly to present methods that may be relevant for CV variational compiling problems in more complicated, large scale settings.

Real spectra of large real asymmetric random matrices.

When a randomness is introduced at the level of real matrix elements, depending on its particular realization, a pair of eigenvalues can appear as real or form a complex conjugate pair. We show that in the limit of large matrix size the density of such real eigenvalues is proportional to the square root of the asymptotic density of complex eigenvalues continuated to the real line. This relation allows one to calculate the real densities up to a normalization constant, which is then applied to various examples, including heavy-tailed ensembles and adjacency matrices of sparse random regular graphs.

Multi-parameter estimation with multi-mode Ramsey interferometry

Estimating multiple parameters simultaneously is of great importance to measurement science and application. For a single parameter, atomic Ramsey interferometry (or equivalently optical Mach–Zehnder interferometry) is capable of providing the precision at the standard quantum limit (SQL) using unentangled probe states as input. In such an interferometer, the first beam splitter represented by unitary transformation U generates a quantum phase sensing superposition state, while the second beam splitter U−1 recombines the phase encoded paths to realize interferometric sensing in terms of population measurements. We prove that such an interferometric scheme can be directly generalized to estimation of multiple parameters (associated with commuting generators) to the SQL precision using multi-mode unentangled states, if (but not iff) U is orthogonal, i.e. a unitary transformation with only real matrix elements. We show that such a U can always be constructed experimentally in a simple and scalable manner. The effects of particle number fluctuation and detection noise on such multi-mode interferometry are considered. Our findings offer a simple solution for estimating multiple parameters corresponding to mutually commuting generators.

Exact Time Evolution of the Asymmetric Hubbard Dimer

We examine the time evolution of an asymmetric Hubbard dimer, which has a different on-site interaction on the two sites. The Hamiltonian has a time-dependent hopping term, which can be employed to describe an electric field (which creates a Hamiltonian with complex matrix elements), or it can describe a modulation of the lattice (which has real matrix elements). By examining the symmetries under spin and pseudospin, we show that the former case involves at most a 3 x 3 block---it can be mapped onto the time evolution of a time-independent Hamiltonian, so the dynamics can be evaluated analytically and exactly (by solving a nontrivial cubic equation). We also show that the latter case reduces to at most 2 x 2 blocks, and hence the time evolution for a single Trotter step can be determined exactly, but the time evolution generically requires a Trotter product.

Non-Hermitian extensions of Wishart random matrix ensembles

We briefly review the solution of three ensembles of non-Hermitian random matrices generalizing the Wishart-Laguerre (also called chiral) ensembles. These generalizations are realized as Gaussian two-matrix models, where the complex eigenvalues of the product of the two independent rectangular matrices are sought, with the matrix elements of both matrices being either real, complex or quaternion real. We also present the more general case depending on a non-Hermiticity parameter, that allows us to interpolate between the corresponding three Hermitian Wishart ensembles with real eigenvalues and the maximally non-Hermitian case. All three symmetry classes are explicitly solved for finite matrix size NxM for all complex eigenvalue correlations functions (and real or mixed correlations for real matrix elements). These are given in terms of the corresponding kernels built from orthogonal or skew-orthogonal Laguerre polynomials in the complex plane. We then present the corresponding three Bessel kernels in the complex plane in the microscopic large-N scaling limit at the origin, both at weak and strong non-Hermiticity with M-N greater or equal to 0 fixed.

Non-aligned hydrogen molecular ion in strong magnetic fields

The hydrogen molecular ion in a strong magnetic field is studied for arbitrary orientations of the molecular axis. A gauge preserving the parity symmetry and leading to real matrix elements for a class of basis states is introduced. The calculations are performed in prolate spheroidal coordinates with the Lagrange-mesh method. The simple resulting mesh equations provide a high accuracy with short computing times for magnetic fields γ = 1 and 10 in atomic units of 2.35 × 105 T. Less accurate results are obtained at the field γ = 100, typical of neutron stars where the size of the matrix becomes very large. The present results allow evaluation of the accuracy of published variational results. At high field strengths, the rotational motion becomes strongly hindered. Simple estimates of rotational energies are compared with other works.

Spin eigenfunctions and operators for the Dn groups

An automated procedure for determining symmetry-adapted spin eigenfunctions is developed for the D n symmetry groups with n equivalent spins for arbitrary size of I and for n=3,…,6. These eigenfunctions are also eigenfunctions of the vector sum of the n equivalent nuclear spins. Generalized hyperfine spin operators are developed that have real matrix elements and that exploit the full symmetry. These spin operators can be combined with operators for other spins and molecular rotation using only real arithmetic.

Why two qubits are special

We analyze some special properties of a system of two qubits, and in particular of the so-called Bell basis for this system, and discuss the possibility of extending these properties to higher dimensional systems. We give a general construction for orthonormal bases of maximally entangled vectors, which works in any dimension, and is based on Latin squares and complex Hadamard matrices. However, for none of these bases the special properties of the operation of complex conjugation in Bell basis hold, namely that maximally entangled vectors have up-to-a-phase real coefficients and that factorizable unitaries have real matrix elements.

Classical Hamiltonian description of a two-level quantum system: The quantum Zeno effect

We consider the simplest quantum system, namely the two-level system with real matrix elements, and exhibit its specific phase properties related to the complex Hamiltonian formalism provided by a simple canonical transformation between the Schroedinger equation and a classical dynamical system that actually describes the dynamics of a spin 1 2 . In the case of the quantum Zeno effect, it yields the provocative theoretical suggestion and experimental prediction that an energy measurement should always yield the same energy value (which is either the upper or the lower energy level, depending on the value of the energy expectation value), contrary to the standard quantum rule stating that an energy measurement should share the probabilities between the two energy levels (except for the eigenstates). This makes a quasi-continuously observed two-level quantum system very similar to a classical system from the point of view of energy measurements.

Random matrix triality at nonzero chemical potential

We introduce three universality classes of chiral random matrix ensembles with a nonzero chemical potential and real, complex or quaternion real matrix elements. In the thermodynamic limit we find that the distribution of the eigenvalues in the complex plane does not depend on the Dyson index, and is given by the solution proposed by Stephanov. For a finite number of degrees of freedom, $N,$ we find an accumulation of eigenvalues on the imaginary axis for real matrices, whereas for quaternion real matrices we find a depletion of eigenvalues in this domain. This effect is of order $1/\sqrt{N}.$ In particular for the real case the resolvent shows a discontinuity of order $1/\sqrt{N}.$ These results are in agreement with lattice QCD simulations with staggered fermions and recent instanton liquid simulations both for two colors and a nonzero chemical potential.

Influence of Collective Decay of Excited Electronic States of Chromophores on Dispersion Characteristics of Resonance Raman Scattering by Chromophore Dimers with Dynamic Coupling Between Chromophores

Expressions for the excitation profiles and dispersion curves of the depolarization ratio of Raman lines and for the Raman optical activity signal were obtained for freely orienting rigid chromophore dimers, for which the interactions between identical chromophores are due to both ‘collective decay’ of the excited electronic states of the chromophores (described by off-diagonal elements of the dimer damping matrix) and ‘dynamic coupling’ (described by a coupling operator with real matrix elements). The frequency region near the frequency of the pure electronic transition of the chromophore is considered. It is shown that in such chromophores the collective decay leads to suppression and widening one of two components of excitation profile of Raman line (separated by dynamic coupling) and to enhancement and narrowing the other. There is the same influence on intensity and width of two components of dispersion curves of Raman optical activity signal. The collective decay also causes shifting of the extremes of the dispersion curves of the depolarization ratio of the Raman line towards the enhancing components of the excitation profiles and a decrease in width, when the vectors of the transition dipoles of the chromophores are not parallel or antiparallel. Calculated examples are presented.

Spectrum of the Dirac operator in a QCD instanton liquid: two versus three colors

Approximating the sum over all gauge field configurations in the QCD partition function by a liquid of instantons, we calculate the spectrum of the Dirac operator for two and three colors and for zero, one and two flavors. We find a remarkable difference in the spectrum near zero virtuality between two and three colors, which can be explained in terms of chiral random matrix theory. For two colors the Dirac operator is real, and the appropriate random matrix ensemble has real matrix elements. For three colors the Dirac operator is complex, and the spectrum can be described by a random matrix ensemble with complex matrix elements. These results provide further evidence that the spectrum near zero virtuality is universal and is completely determined by symmetries.

The spectrum of the Dirac operator near zero virtuality for Nc = 2 and chiral random matrix theory

We study the spectrum of the QCD Dirac operator near zero virtuality for N c = 2. According to a universality argument, it can be described by a random matrix theory with the chiral structure of QCD, but with real matrix elements. Using results derived by Mehta and Mahoux and Nagao and Wadati, we are able to obtain an analytical result for the microscopic spectral density that in turn is the generating function for Leutwyler-Smilga type spectral sum rules.

Higher-order corrections to jet cross sections in e+e- annihilation.

A general method to calculate next-to-leading order multijet cross sections is presented. The emphasis is on how to isolate the soft and collinear divergences in multiparton matrix elements. As an example, the method is used to isolate the divergences at leading order in the number of colours in the processes e+e- + qQ + n gluons and e+e- -t qcjqQ + n gluons. The usual algebraic complexity of calculating next-to-leading order corrections in QCD is avoided, especially the d-dimensional squaring of the real matrix elements and the hard phase space integrals. Some remarks about the stucture of the virtual contributions are made. As a first application, and to examine the feasibility of the approach, explicit Monte Carlo programs are constructed which contain the next-to-leading order corrections to e+e- -+ 2 jets and e+e- + 3 jets. It is demonstrated that the method works and can be readily applied to a variety of processes.