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Physical Review A

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

Jerzy Ciosłowski

Published Oct 1, 1987• 19 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.36.3441

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Further advancement of quantum computing (QC) is contingent on enabling many-body models that avoid deep circuits and excessive use of CNOT gates. To this end, we develop a QC approach employing finite-order connected moment expansions (CMX) and affordable procedures for initial state preparation. We demonstrate the performance of our approach employing several quantum variants of CMX through the classical emulations on the H2 molecule potential energy surface and the Anderson model with a broad range of correlation strength. The results show that our approach is robust and flexible. Good agreement with exact solutions can be maintained even at the dissociation and strong correlation limits.

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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.
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.
The ttt-expansion method for calculating ground-state energies of Hamiltonian systems is generalized to excited states. We introduce quantities Mn(t){M}_{n}(t)Mn​(t) and Dn(t){D}_{n}(t)Dn​(t) which for $t\ensuremath{\rightarrow}\ensuremath{\infty}$ converge to ${\ensuremath{\Sigma}}_{n}={E}_{0}+{E}_{1}+\dots{}+{E}_{n}$ and ${\ensuremath{\Delta}}_{n}={E}_{n+1}\ensuremath{-}{E}_{n}$, respectively. Like in the standard ttt-expansion, the ttt-series for Mn(t){M}_{n}(t)Mn​(t) and Dn(t){D}_{n}(t)Dn​(t) are constructed from moments of the Hamiltonian. The values of ${\ensuremath{\Sigma}}_{n}$ and ${\ensuremath{\Delta}}_{n}$ can then be estimated using (e.g.) a generalization of the connected-moment expansion of Cioslowski.
Further advancement of quantum computing (QC) is contingent on enabling many-body models that avoid deep circuits and excessive use of CNOT gates. To this end, we develop a QC approach employing finite-order connected moment expansions (CMX) and affordable procedures for initial state preparation. We demonstrate the performance of our approach employing several quantum variants of CMX through the classical emulations on the H2 molecule potential energy surface and the Anderson model with a broad range of correlation strength. The results show that our approach is robust and flexible. Good agreement with exact solutions can be maintained even at the dissociation and strong correlation limits.