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Progress in Nuclear Science and Technology

Rapid Scheme of Producing Generalized Fourier Expansion of Matrix Functions and its Application to Physical Problems

Toma MATSUMOTO, Yoko Yamane, Atsunori TANAKA, Shigetoshi Sota, Masaki Itoh

Published Oct 1, 2011• 0 citations
Publisher

Abstract

Application of orthogonal polynomial expansion to quantum simulations is formulated on a general footing, implementing the regulation technique by Sota and Itoh for treating for the Gibbs oscillation.It is an alternative to the kernel polynomial method using Tchebyshev polynomial, but is simpler to handle and makes it possible to use all the popular orthogonal polynomials, covering finite, semi-infinite and infinite intervals of the eigenvalue spectrum.The accuracy can be made equivalent to direct diagonalization, with the resolution being homogeneous in the whole range of the spectrum.The target quantities can be as diverse as including eigenvectors, as well as all sorts of one-particle properties and correlation functions, involving thermal average and quantum time evolution.It can also be used as a handy tool for solving linear algebraic equations.

DOI
10.15669/pnst.2.688

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Application of orthogonal polynomial expansion to quantum simulations is formulated on a general footing, implementing the regulation technique by Sota and Itoh for treating for the Gibbs oscillation.It is an alternative to the kernel polynomial method using Tchebyshev polynomial, but is simpler to handle and makes it possible to use all the popular orthogonal polynomials, covering finite, semi-infinite and infinite intervals of the eigenvalue spectrum.The accuracy can be made equivalent to direct diagonalization, with the resolution being homogeneous in the whole range of the spectrum.The target quantities can be as diverse as including eigenvectors, as well as all sorts of one-particle properties and correlation functions, involving thermal average and quantum time evolution.It can also be used as a handy tool for solving linear algebraic equations.
An oscillation-free Fourier expansion is proposed and applied to the polynomial expansion of resolvent operator, as an alternative treatment for Gibbs phenomenon to the kernel polynomial method (KPM) by Voter et al. [Phys. Rev. B 53 (1996) 12733]. Adopting regulated Legendre polynomial as basis functions, the recurrence algorithm underlying the KPM is practically preserved, while it eliminates unphysical oscillations entirely. The resolution is uniform in the frequency domain, and improved without limitation by extending the truncation range of the expansion. The method can also be used to calculate the eigenvector with remarkably high precision. It is further extended to the time domain, being able to deal with the quantum evolution in the same algorithm. A numerical test was performed against a lattice dynamics of 19 3 atoms, as well as of a computer generated Lennard–Jones glass of 1000 atoms. Not only it serves as a most efficient simulation scheme for a bulk, confirmed was also that the calculation converges to exact results within realistic computation time.
Efficient and stable algorithms for the calculation of spectral quantities and correlation functions are some of the key tools in computational condensed-matter physics. In this paper basic properties and recent developments of Chebyshev expansion based algorithms and the kernel polynomial method are reviewed. Characterized by a resource consumption that scales linearly with the problem dimension these methods enjoyed growing popularity over the last decade and found broad application not only in physics. Representative examples from the fields of disordered systems, strongly correlated electrons, electron-phonon interaction, and quantum spin systems are discussed in detail. In addition, an illustration on how the kernel polynomial method is successfully embedded into other numerical techniques, such as cluster perturbation theory or Monte Carlo simulation, is provided.
We present a formulation of quantum mechanics based on orthogonal polynomials. The wavefunction is expanded over a complete set of square integrable basis in configuration space where the expansion coefficients are orthogonal polynomials in the energy. Information about the corresponding physical systems (both structural and dynamical) are derived from the properties of these polynomials. We demonstrate that an advantage of this formulation is that the class of exactly solvable non-relativistic quantum mechanical problems becomes larger than in the conventional formulation (see, for example, Table 1 in the text).
Computing finite temperature properties of a quantum many-body system is key to describing a broad range of correlated quantum many-body physics from quantum chemistry and condensed matter to thermal quantum field theories. Quantum computing, which has seen rapid developments in recent years, has a huge potential to impact the computation of quantum thermodynamics. To fulfill these potential impacts, it is crucial to design quantum algorithms that utilize the computation power of quantum computing devices. Here, we present a quantum kernel function expansion (QKFE) algorithm for solving thermodynamic properties of quantum many-body systems. In this quantum algorithm, the many-body density of states is approximated by a kernel Fourier expansion, whose expansion moments are obtained by random state sampling and quantum interferometric measurements. Compared to its classical counterpart, namely, the kernel polynomial method (KPM), QKFE has an exponential advantage in the cost of both time and memory. In computing low temperature properties, QKFE becomes inefficient, similar to classical KPM. To resolve this difficulty, we further construct a thermal ensemble iteration (THEI) protocol which starts from the trivial limit of an infinite temperature ensemble and approaches the low temperature regime step by step. For quantum Hamiltonians, whose ground states are preparable with polynomial quantum circuits, THEI has an overall polynomial complexity. We demonstrate its efficiency with applications to one- and two-dimensional quantum spin models and a fermionic lattice. With our analysis of the realization with digital and analog quantum devices, we believe the quantum algorithm is accessible to current quantum technology.
The application of the quantum Fourier transform (QFT) within the field of quantum computation has been manifold. Shor’s algorithm, phase estimation and computing discrete logarithms are but a few classic examples of its use. These initial blueprints for quantum algorithms have sparked a cascade of tantalizing solutions to problems considered to be intractable on a classical computer. Therefore, two main threads of research have unfolded. First, novel applications and algorithms involving the QFT are continually being developed. Second, improvements in the algorithmic complexity of the QFT are also a sought after commodity. In this work, we review the structure of the QFT and its implementation. In order to put these concepts in their proper perspective, we provide a brief overview of quantum computation. Finally, we provide a permutation structure for putting the QFT within the context of universal computation.
Abstract Quantum models of interacting bosons have a wide range of applications, including the propagation of optical modes in nonlinear media, such as the k-photon down-conversion. Many of these models are related to nonlinear deformations of finite group algebras and, in this sense, are exactly solvable. While advanced group-theoretic methods were developed to study the eigenvalue spectrum, in quantum optics, the primary focus is not on the spectrum of the Hamiltonian but rather on the evolution of an initial state—such as the generation of optical signal modes by a strong pump mode propagating through a nonlinear medium. I propose a simple and general method to solve the state evolution problem, applicable to a broad class of quantum models of interacting bosons. For the k-photon down-conversion model and its generalizations, the solution to the state evolution problem is expressed as an infinite series expansion in powers of the propagation time, with coefficients determined by a recursion relation involving only a single polynomial function. This polynomial function is unique to each nonlinear model. As an application, I compare the exact solution of the parametric down-conversion process with the semiclassical parametric approximation.