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Physical Review Letters

Connected moments expansion: A new tool for quantum many-body theory

Jerzy Ciosłowski

Published Jan 12, 1987• 129 citations
Publisher

Abstract

The connected-moments expansion (CMX) for the ground-state energy is derived from the Horn-Weinstein theorem. This new technique possesses several striking advantages over traditional approaches to calculation of the ground-state energy. Like perturbational theories, the CMX series is conceptually simple but is easier to derive and program and is usually convergent even for very ``crude'' trial kets. Numerical examples reveal potential applicability of CMX for calculation of the correlation energy when the trial ket has the form of multideterminant wave function.

DOI
10.1103/physrevlett.58.83

Related papers

Estimation of the overlap between the approximate and exact wave function of the ground state from the connected-moments expansion

Jerzy Ciosłowski

A connected-moments expansion (CMX) is constructed for the overlap between the approximate and the exact (but unknown) wave function of the ground state. The CMX series provides approximations to the overlap that do not require the knowledge of the exact energy of either the ground or the first excited state. Moreover, the obtained approximate overlaps have a correct size extensivity. A numerical example is given, demonstrating the performance of the new method.

Physical Review A•Oct 1, 1987•19 citations

Analytic properties of connected moments expansions

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Abstract The derivation of the connected moments expansion (CMX) is examined as well as the singularities that arise in the series expansion for the ground‐state energy. Explicit analytic results are presented that show a canceling of these singularities. Also, an alternate moments expansion (AMX) is derived that closely models the CMX but displays a varied computational range. © 1994 John Wiley & Sons, Inc.

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Prony's method and the connected-moments expansion

Francisco Marcelo Fernández

We show that Prony's method provides the full solution to the nonlinear equations of the connected--moments expansion (CMX). Knowledge of all the parameters in the CMX ansatz is useful for the analysis of the convergence properties of the approach. Prony's method is also suitable for the calculation of the correlation function for simple quantum--mechanical models.

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Prony's method and the connected-moments expansion

Francisco Marcelo Fernández

We show that Prony's method provides the full solution to the nonlinear equations of the connected-moments expansion (CMX). Knowledge of all the parameters in the CMX ansatz is useful for the analysis of the convergence properties of the approach. Prony's method may also be suitable for the calculation of the correlation function for simple quantum-mechanical models.

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Improving the accuracy and efficiency of quantum connected moments expansions *

Daniel Claudino, Bo Peng, Nicholas P Bauman +2 more

Abstract The still-maturing noisy intermediate-scale quantum technology faces strict limitations on the algorithms that can be implemented efficiently. In the realm of quantum chemistry, the variational quantum eigensolver (VQE) algorithm has become ubiquitous, with many variations. Alternatively, a promising new avenue has been unraveled by the quantum variants of techniques grounded on expansions of the moments of the Hamiltonian, notably the connected moments expansion (CMX) and the Peeters–Devreese–Soldatov (PDS) energy functional. Common to those approaches is that, upon preparing an approximate ground state used to compute the necessary moments, the accuracy of the estimated ground state energy depends on the degree of overlap between the prepared state and the true ground state. Thus, we use the ADAPT-VQE algorithm to test shallow circuit construction strategies for the purpose of increasing the overlap with the exact ground state, validated by the sizable accuracy improvement herein reported in the PDS and CMX ground state energies. We also show that we can take advantage of the fact that the terms to be measured are highly recurring in different moments, incurring a substantial reduction in the number of necessary measurements. By coupling this measurement caching with a threshold that determines whether a given term is to be measured based on its associated scalar coefficient, we observe a further reduction in the number of circuit implementations while allowing for tunable accuracy.

arXiv:2103.09124•Quantum Science and Technology•May 18, 2021•26 citations

Connected moments expansion for the ground-state energy of systems described by nonlinear Hamiltonians

Jerzy Ciosłowski

An extension of the connected moments expansion to nonlinear Hamiltonians is presented. The new method enables construction of rapidly convergent series for the ground-state energy which requires a relatively small amount of numerical effort.

Physical Review A•Jul 1, 1987•37 citations
The connected-moments expansion (CMX) for the ground-state energy is derived from the Horn-Weinstein theorem. This new technique possesses several striking advantages over traditional approaches to calculation of the ground-state energy. Like perturbational theories, the CMX series is conceptually simple but is easier to derive and program and is usually convergent even for very ``crude'' trial kets. Numerical examples reveal potential applicability of CMX for calculation of the correlation energy when the trial ket has the form of multideterminant wave function.
Abstract The still-maturing noisy intermediate-scale quantum technology faces strict limitations on the algorithms that can be implemented efficiently. In the realm of quantum chemistry, the variational quantum eigensolver (VQE) algorithm has become ubiquitous, with many variations. Alternatively, a promising new avenue has been unraveled by the quantum variants of techniques grounded on expansions of the moments of the Hamiltonian, notably the connected moments expansion (CMX) and the Peeters–Devreese–Soldatov (PDS) energy functional. Common to those approaches is that, upon preparing an approximate ground state used to compute the necessary moments, the accuracy of the estimated ground state energy depends on the degree of overlap between the prepared state and the true ground state. Thus, we use the ADAPT-VQE algorithm to test shallow circuit construction strategies for the purpose of increasing the overlap with the exact ground state, validated by the sizable accuracy improvement herein reported in the PDS and CMX ground state energies. We also show that we can take advantage of the fact that the terms to be measured are highly recurring in different moments, incurring a substantial reduction in the number of necessary measurements. By coupling this measurement caching with a threshold that determines whether a given term is to be measured based on its associated scalar coefficient, we observe a further reduction in the number of circuit implementations while allowing for tunable accuracy.