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arXiv:1112.0292

The Rayleigh Quotient

Michael Kruse, Justin M. Conroy, H. G. Miller

Published Dec 1, 2011• 6 citations
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Abstract

The central role of the Rayleigh quotient in many body physics is discussed. Various many body methods can be obtained from either an attempt to evaluate the Rayleigh Quotient directly or through various variational approximations. Rather than dwell on the technical details necessary to obtain the equations of the various many body methods, we concentrate on how they can be obtained from the Rayleigh Quotient, and some of the consequences of the approximations involved in their evaluation.

DOI
10.48550/arxiv.1112.0292

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The central role of the Rayleigh quotient in many body physics is discussed. Various many body methods can be obtained from either an attempt to evaluate the Rayleigh Quotient directly or through various variational approximations. Rather than dwell on the technical details necessary to obtain the equations of the various many body methods, we concentrate on how they can be obtained from the Rayleigh Quotient, and some of the consequences of the approximations involved in their evaluation.
ABSTRACT The analysis of natural frequency is an important problem of structural engineering, where complex structures often lead to expensive eigenvalue problems. Variational quantum eigensolvers provide a potentially efficient solution by using quantum computing, but their application is limited by the costly need to decompose system matrices into a linear combination of unitaries. In this work, we apply a recently proposed decomposition‐free method from linear systems to eigenvalue problems in structural engineering. Specifically, we develop two algorithms based on this method, the decomposition‐free variational quantum eigensolver (DF‐VQE) and generalized eigensolver (DF‐VQGE) for the standard eigenvalue problems and the generalized eigenvalue problems, respectively. In these algorithms, the Rayleigh Quotient and generalized Rayleigh Quotient cost evaluations are reformulated so that the need for matrix decomposition is bypassed. The central technical step in this application is to use vectorization to encode the system matrices directly into quantum state vectors, allowing their properties to be computed via a standard swap test quantum circuit. The proposed methods are validated on a quantum simulator with three representative structural engineering problems, including a 1D bar, a 2D beam, and a 3D truss bridge, obtaining results that align well with reference solutions from classical finite element analysis. We also discuss the quantum state preparation challenge that must be addressed before this formulation can be connected to meaningful quantum advantage.
We introduce Rayleigh functional for nonlinear systems. It is defined using the energy functional and the normalization properties of the variables of variation. The key property of the Rayleigh quotient for linear systems is preserved in our definition: the extremals of the Rayleigh functional coincide with the stationary solutions of the Euler-Lagrange equation. Moreover, the second variation of the Rayleigh functional defines stability of the solution. This gives rise to a powerful numerical optimization method in the search for the energy minimizers. It is shown that the well-known imaginary time relaxation is a special case of our method. To illustrate the method we find the stationary states of Bose-Einstein condensates in various geometries. Finally, we show that the Rayleigh functional also provides a simple way to derive analytical identities satisfied by the stationary solutions of the critical nonlinear equations. We also show that the functional used in Phys. Rev. A 66, 036612 (2002) is erroneous.
Both single-well and multiwell one-dimensional anharmonic oscillators have provided an extremely fruitful testing ground for various modern techniques of microscopic quantum many-body theory and quantum field theory. They have served both as (0+1)-dimensional field-theory models of their important counterparts in a higher number of dimensions, and as algebraically simple yet highly nontrivial problems on which various approximation or truncation schemes could be tested to very high orders. Such techniques have included perturbation theory and its many possible resummation schemes, and the coupled-cluster method. In the present paper the spotlight is focused on variational methods, and we concentrate on their application to the particularly demanding double-well quartic anharmonic oscillator. We examine in detail both single-state and multistate variational calculations based on either orthogonal (uncorrelated) or nonorthogonal (correlated) basis functions. Particular attention is paid to the question of the sensitivity and accuracy of the results with respect to any nonlinear parameters that characterize the basis states. Practical schemes are suggested for their choice and are shown to be capable of very high accuracy. Use of this simple model also allows detailed comparisons to be made with other techniques of a perturbative or perturbative-variational hybrid character.