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arXiv:2211.00790 · The European Physical Journal A

Faster spectral density calculation using energy moments

Jeremy Hartse, Alessandro Roggero

Published Mar 9, 2023• 9 citations
arXivHTMLPDF

Abstract

Abstract Accurate predictions of inclusive scattering cross sections in the linear response regime require efficient and controllable methods to calculate the spectral density in a strongly-correlated many-body system. In this work we reformulate the recently proposed Gaussian Integral Transform technique in terms of Fourier moments of the system Hamiltonian which can be computed efficiently on a quantum computer. One of the main advantages of this framework is that it allows for an important reduction of the computational cost by exploiting previous knowledge about the energy moments of the spectral density. For a simple model of medium mass nucleus like $$^{40}$$ 40 Ca and target energy resolution of 1 MeV we find an expected speed-up of $$\approx 125$$ ≈ 125 times for the calculation of the giant dipole response and of $$\approx 50$$ ≈ 50 times for the simulation of quasi-elastic electron scattering at typical momentum transfers.

DOI
10.1140/epja/s10050-023-00952-6

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Abstract Accurate predictions of inclusive scattering cross sections in the linear response regime require efficient and controllable methods to calculate the spectral density in a strongly-correlated many-body system. In this work we reformulate the recently proposed Gaussian Integral Transform technique in terms of Fourier moments of the system Hamiltonian which can be computed efficiently on a quantum computer. One of the main advantages of this framework is that it allows for an important reduction of the computational cost by exploiting previous knowledge about the energy moments of the spectral density. For a simple model of medium mass nucleus like 40^{40}40 40 Ca and target energy resolution of 1 MeV we find an expected speed-up of ≈125\approx 125≈125 ≈ 125 times for the calculation of the giant dipole response and of ≈50\approx 50≈50 ≈ 50 times for the simulation of quasi-elastic electron scattering at typical momentum transfers.
Accurate calculations of the spectral density in a strongly correlated quantum many-body system are of fundamental importance to study its dynamics in the linear response regime. Typical examples are the calculation of inclusive and semiexclusive scattering cross sections in atomic nuclei and transport properties of nuclear and neutron star matter. Integral transform techniques play an important role in accessing the spectral density in a variety of nuclear systems. However, their accuracy is in practice limited by the need to perform a numerical inversion which is often ill-conditioned. In the present work we extend a recently proposed quantum algorithm which circumvents this problem. We show how to perform controllable reconstructions of the spectral density over a finite energy resolution with rigorous error estimates. An appropriate expansion in Chebyshev polynomials allows for efficient simulations also on classical computers. We apply our idea to obtain the local density of states for graphene in a magnetic field as a proof of principle. This paves the way for future applications in nuclear and condensed matter physics.
The spectral-density operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}(\ensuremath{\omega})=\ensuremath{\delta}(\ensuremath{\omega}\ensuremath{-}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H})$ plays a central role in linear response theory as its expectation value, the dynamical response function, can be used to compute scattering cross sections. In this work, we describe a near optimal quantum algorithm providing an approximation to the spectral density with energy resolution $\mathrm{\ensuremath{\Delta}}$ and error $\ensuremath{\epsilon}$ using $O\left(\sqrt{{log}_{2}\left(1/\ensuremath{\epsilon}\right)\left[{log}_{2}\left(1/\mathrm{\ensuremath{\Delta}}\right)+{log}_{2}\left(1/\ensuremath{\epsilon}\right)\right]}/\mathrm{\ensuremath{\Delta}}\right)$ operations. This is achieved without using expensive approximations to the time-evolution operator, but instead exploiting qubitization to implement an approximate Gaussian integral transform of the spectral density. We also describe appropriate error metrics to assess the quality of the spectral function approximations more generally.
While new light sources allow for unprecedented resolution in experiments with X-rays, a theoretical understanding of the scattering cross-section is lacking. In the particular case of strongly correlated electron systems, numerical techniques are quite limited, since conventional approaches rely on calculating a response function (Kramers-Heisenberg formula) that is obtained from a perturbative analysis of scattering processes in the frequency domain. This requires a knowledge of a full set of eigenstates in order to account for all intermediate processes away from equilibrium, limiting the applicability to small tractable systems. In this work, we present an alternative paradigm, recasting the problem in the time domain and explicitly solving the time-dependent Schrödinger equation without the limitations of perturbation theory: a faithful simulation of the scattering processes taking place in actual experiments, including photons and core electrons. We show how this approach can yield the full time and momentum resolved Resonant Inelastic X-Ray Scattering (RIXS) spectrum of strongly interacting many-body systems. We demonstrate the formalism with an application to Mott insulating Hubbard chains using the time-dependent density matrix renormalization group method, which does not require a priory knowledge of the eigenstates and can solve very large systems with dozens of orbitals. This approach can readily be applied to systems out of equilibrium without modification and generalized to other spectroscopies.
Abstract While new light sources allow for unprecedented resolution in experiments with X-rays, a theoretical understanding of the scattering cross-section is lacking. In the particular case of strongly correlated electron systems, numerical techniques are quite limited, since conventional approaches rely on calculating a response functon(Kramers-Heisenberg formula) that is obtained from a perturbative analysis of scattering processes in the frequency domain. This requires a knowledge of a full set of eigenstates in order to account for all intermediate processes away from equilibrium, limiting the applicability to small tractable systems. In this work, we present an alternative paradigm, recasting the problem in the time domain and explicitly solving the time-dependent Schr\"odinger equation without the limitations of perturbation theory: a faithful simulation of the scattering processes taking place in actual experiments, including photons and core electrons. We show how this approach can yield the full time and momentum resolved Resonant Inelastic X-Ray Scattering (RIXS) spectrum of strongly interacting many-body systems. We demonstrate the formalism with an application to Mott insulating Hubbard chains using the time-dependent density matrix renormalization group method, which does not require a priory knowledge of the eigenstates and can solve very large systems with dozens of orbitals. This approach can readily be applied to systems out of equilibrium without modification and generalized to other spectroscopies.
We compute nuclear response functions by solving the time-dependent A-body Schrödinger equation, recording the time-dependent transition moment and extracting spectral information via Fourier transforms. The solution of the time-dependent many-body problem accounts for correlations on top of the mean field by taking advantage of a time-dependent formulation of coupled-cluster theory. As a validation, we focus on electric dipole transitions in 4^44He and 16^{16}16O and compare moments of the response function distribution to the results of an equivalent static framework, finding negligible discrepancies. We investigate how proton and neutron densities evolve in time, and we see the traditional picture of soft and giant dipole resonances as collective oscillations of protons and neutrons emerging from our calculations in 16^{16}16O and 24^{24}24O. This method also allows us to investigate the behavior of the nucleus in the presence of a strong electric field. In that regime, the behavior of the system becomes chaotic. Qualitatively, the spectral information obtained in this limit is in line with previous time-dependent mean-field results.
We present an efficient algorithm for calculating spectral properties of large sparse Hamiltonian matrices such as densities of states and spectral functions. The combination of Chebyshev recursion and maximum entropy achieves high-energy resolution without significant roundoff error, machine precision, or numerical instability limitations. If controlled statistical or systematic errors are acceptable, CPU and memory requirements scale linearly in the number of states. The inference of spectral properties from moments is much better conditioned for Chebyshev moments than for power moments. We adapt concepts from the kernel polynomial method, a linear Chebyshev approximation with optimized Gibbs damping, to control the accuracy of Fourier integrals of positive nonanalytic functions. We compare the performance of kernel polynomial and maximum entropy algorithms for an electronic structure example.