Spectral density estimation with the Gaussian Integral Transform

Abstract

The spectral density operator $\hat{\rho }(\omega )=\delta (\omega -\hat{H})$ 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 $\Delta$ and error $\epsilon$ using $\mathcal{O}\left(\sqrt{\log \left(1/\epsilon \right)\left(\log \left(1/\Delta \right)+\log \left(1/\epsilon \right)\right)}/\Delta \right)$ operations. This is achieved without using expensive approximations to the time-evolution operator but exploiting instead qubitization to implement an approximate Gaussian Integral Transform (GIT) of the spectral density. We also describe appropriate error metrics to assess the quality of spectral function approximations more generally.

arXiv comment12 pages - published version

References in corpus (26)

• Quantum Computing in the NISQ era and beyond
• Quantum Algorithms Revisited
• Hamiltonian Simulation by Qubitization
• Sinc approximation of algebraically decaying functions
• Optimal Hamiltonian Simulation by Quantum Signal Processing
• Simulating Hamiltonian dynamics with a truncated Taylor series
• Quantum algorithm for systems of linear equations with exponentially improved dependence on precision
• Toward the first quantum simulation with quantum speedup
• Optimal states and almost optimal adaptive measurements for quantum interferometry
• Response Functions from Integral Transforms with a Lorentz Kernel
• Optimal Quantum Measurements of Expectation Values of Observables
• Efficient Bayesian Phase Estimation
• Quantum phase estimation of multiple eigenvalues for small-scale (noisy) experiments
• The Lorentz Integral Transform (LIT) method and its applications to perturbation induced reactions
• Simulation of Nonequilibrium Dynamics on a Quantum Computer
• Linear Response on a Quantum Computer
• Quantum Computing for Neutrino-nucleus Scattering
• Quantum Simulation of Chemistry with Sublinear Scaling in Basis Size
• First principles description of the giant dipole resonance in 16O
• Electric dipole polarizability from first principles calculations
• New inversion methods for the Lorentz Integral Transform
• Quantum Algorithms for Estimating Physical Quantities using Block-Encodings
• Thermal conductivity and impurity scattering in the accreting neutron star crust
• The Lorentz Integral Transform and its Inversion
• Implementing smooth functions of a Hermitian matrix on a quantum computer
• On the ill-posed character of the Lorentz integral transform

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