The Shanks transform as the exact filter for harmonic inversion

2003 ◽

Vol 02 (04) ◽

pp. 497-505 ◽

Cited By ~ 10

Author(s):

VLADIMIR A. MANDELSHTAM

Keyword(s):

Correlation Functions ◽

Cross Correlation ◽

Window Size ◽

Initial State ◽

Filter Diagonalization ◽

Molecular Systems ◽

Diagonalization Method ◽

Low Dimensional ◽

Harmonic Inversion ◽

Filter Diagonalization Method

Harmonic inversion of Chebyshev correlation and cross-correlation functions by the filter diagonalization method (FDM) is one of the most efficient ways to accurately compute the complex spectra of low dimensional quantum molecular systems. This explains the growing popularity of the FDM in the past several years. Some of its most attractive features are the predictable convergence properties and the lack of adjusting parameters. These issues however are often misunderstood and mystified. We discuss the questions relevant to the optimal choices for the FDM parameters, such as the window size and the number of basis functions. We also demonstrate that the cross-correlation approach (using multiple initial states) is significantly more effective than the conventional autocorrelation approach (single initial state) for the common case of a non-uniform eigenvalue distribution.

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The Shanks transform as the exact filter for harmonic inversion

Series in Atomic Molecular Physics - Quantum-Mechanical Signal Processing and Spectral Analysis ◽

10.1201/9781420033601.ch19 ◽

2004 ◽

pp. 133-149

Keyword(s):

Harmonic Inversion

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A Comparison of Three Ways to Measure Time-Dependent Densities With Quantum Simulators

Frontiers in Physics ◽

10.3389/fphy.2021.546538 ◽

2021 ◽

Vol 9 ◽

Author(s):

Jun Yang ◽

James Brown ◽

James Daniel Whitfield

Keyword(s):

Density Functional ◽

Quantum Algorithms ◽

Quantum Circuit ◽

Time Dependent ◽

Test Results ◽

Functional Theory ◽

Virtual Device ◽

Starting Point ◽

Intractable Problems ◽

Harmonic Inversion

Quantum algorithms are touted as a way around some classically intractable problems such as the simulation of quantum mechanics. At the end of all quantum algorithms is a quantum measurement whereby classical data is extracted and utilized. In fact, many of the modern hybrid-classical approaches are essentially quantum measurements of states with short quantum circuit descriptions. Here, we compare and examine three methods of extracting the time-dependent one-particle probability density from a quantum simulation: direct Z-measurement, Bayesian phase estimation, and harmonic inversion. We have tested these methods in the context of the potential inversion problem of time-dependent density functional theory. Our test results suggest that direct measurement is the preferable method. We also highlight areas where the other two methods may be useful and report on tests using Rigetti's quantum virtual device. This study provides a starting point for imminent applications of quantum computing.

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