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arXiv:1011.2260 · Journal of Physics A Mathematical and Theoretical

Further analysis of the connected moments expansion

Paolo Amore, Francisco Marcelo Fernández, Martin Rodriguez-Vega

Published Nov 22, 2011• 3 citations
arXivHTMLPDF

Abstract

By means of simple quantum-mechanical models we show that under certain conditions the main assumptions of the connected moments expansion (CMX) are no longer valid. In particular, we consider two-level systems: the harmonic oscillator and the pure quartic oscillator. Although derived from such simple models, we think that the results of this investigation may be of utility in future applications of the approach to realistic problems. We show that a straightforward analysis of the CMX exponential parameters may provide a clear indication of the success of the approach.

DOI
10.1088/1751-8113/44/50/505302

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By means of simple quantum-mechanical models we show that under certain conditions the main assumptions of the connected moments expansion (CMX) are no longer valid. In particular, we consider two-level systems: the harmonic oscillator and the pure quartic oscillator. Although derived from such simple models, we think that the results of this investigation may be of utility in future applications of the approach to realistic problems. We show that a straightforward analysis of the CMX exponential parameters may provide a clear indication of the success of the approach.
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.
Now that we have learned some of the principles which apply in quantum mechanics, we move on to the next more difficult problem of a quantized harmonic oscillator. The goal here is to provide a rigorous application of the polynomial method of solving differential equations on a relatively simple case and to provide some insight into how the Schrödinger equation was first solved [1]. Then we proceed to application in the form of worked examples. We have tried to give sufficient details of this solution to allow a student to follow the derivation with pencil and paper but do not forget to ponder over the spectroscopic applications! This author would agree that it is more important to absorb the main conclusions of this material than to master the derivations. In fact, the highest recommendation of this author is to always ask ‘‘What does this mean?’’ and absorb the conclusions for a future activity called ‘‘thinking’’ rather than just memorizing facts.
Quantum harmonic oscillators coupled through coordinates and momenta, represented by the Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}={\ensuremath{\sum}}_{i=1}^{2}(\frac{{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{i}^{2}}{2{m}_{i}}+\frac{{m}_{i}{\ensuremath{\omega}}_{i}^{2}}{2}{x}_{i}^{2})+{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{\text{int}}$, where the interaction of two oscillators ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{\text{int}}=i{k}_{1}{x}_{1}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{2}+i{k}_{2}{x}_{2}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{1}+{k}_{3}{x}_{1}{x}_{2}\ensuremath{-}{k}_{4}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{1}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{2}$, are found in many applications of quantum optics, nonlinear physics, molecular chemistry, and biophysics. Despite this, there is currently no general solution to the Schr\"odinger equation for such a system. This is especially relevant for quantum entanglement of such a system in quantum optics applications. Here this problem is solved and it is shown that quantum entanglement depends on only one coefficient, $R\ensuremath{\in}(0,1)$, which includes all the parameters of the system under consideration. It has been shown that quantum entanglement can be very large at certain values of this coefficient. The results obtained have a fairly simple analytical form, which facilitates analysis.