An efficient quantum algorithm for generation of ab initio n-th order susceptibilities for non-linear spectroscopies

TL;DR

This paper presents a fault-tolerant quantum algorithm for efficiently computing high-order response functions crucial for non-linear spectroscopy analysis, leveraging perturbation theory and eigenstate filtering.

Contribution

It introduces a novel quantum algorithm that efficiently calculates ab initio n-th order susceptibilities, improving computational cost estimates for non-linear spectroscopic properties.

Findings

Algorithm saturates the Heisenberg limit for energy estimation.

Provides cost estimates in terms of block encoding queries.

Enables approximation of transition dipole moments.

Abstract

We develop and analyze a fault-tolerant quantum algorithm for computing $n$-th order response properties necessary for analysis of non-linear spectroscopies of molecular and condensed phase systems. We use a semi-classical description in which the electronic degrees of freedom are treated quantum mechanically and the light is treated as a classical field. The algorithm we present can be viewed as an implementation of standard perturbation theory techniques, focused on {\it ab initio} calculation of $n$-th order response functions. We provide cost estimates in terms of the number of queries to the block encoding of the unperturbed Hamiltonian, as well as the block encodings of the perturbing dipole operators. Using the technique of eigenstate filtering, we provide an algorithm to extract excitation energies to resolution $\gamma$, and the corresponding linear response amplitude to accuracy $\epsilon$ using ${O}\left(N^{6}\eta^2{{\gamma^{-1}}\epsilon^{-1}}\log(1/\epsilon)\right)$ queries to the block encoding of the unperturbed Hamiltonian $H_0$, in double factorized representation. Thus, our approach saturates the Heisenberg $O(\gamma^{-1})$ limit for energy estimation and allows for the approximation of relevant transition dipole moments. These quantities, combined with sum-over-states formulation of polarizabilities, can be used to compute the $n$-th order susceptibilities and response functions for non-linear spectroscopies under limited assumptions using $\widetilde{O}\left({N^{5n+1}\eta^{n+1}}/{\gamma^n\epsilon}\right)$ queries to the block encoding of $H_0$.