We develop and analyze a fault-tolerant quantum algorithm for non-linear response properties of molecular and condensed phase systems. We consider a semi-classical description in which the electronic degrees of freedom are treated quantum mechanically and the light is treated as a classical field interacting via the electric dipole approximation. We use the technique of eigenstate filtering, to efficiently resolve excitation energies for dominant dipole transitions. When applied to the electronic structure Hamiltonian with double factorized representation, each significant spectral line can be approximated to a width of ±γ, and to a height of ±ϵ with <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41534_2025_1026_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="177" /> </InlineMediaObject> <EquationSource Format="TEX">$O\left({N}^{6}{\eta }^{2}{\gamma }^{-1}{\epsilon }^{-1}\log (1/\epsilon )\right)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mfenced close=")" open="("> <mrow> <msup> <mrow> <mi>N</mi> </mrow> <mrow> <mn>6</mn> </mrow> </msup> <msup> <mrow> <mi>η</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mi>γ</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mi>ϵ</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>log</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>/</mo> <mi>ϵ</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfenced> </mrow> </math></EquationSource> </InlineEquation> queries to the block encoding of the unperturbed electronic structure Hamiltonian for η electrons over N basis functions. These quantities can be used to compute the nth order response functions for non-linear spectroscopies under limited assumptions using <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41534_2025_1026_Article_IEq2.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="141" /> </InlineMediaObject> <EquationSource Format="TEX">$\widetilde{O}\left({N}^{5n+1}{\eta }^{n+1}/{\gamma }^{n}\epsilon \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mrow> <mi>O</mi> </mrow> <mo>̃</mo> </mover> <mfenced close=")" open="("> <mrow> <msup> <mrow> <mi>N</mi> </mrow> <mrow> <mn>5</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mi>η</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>/</mo> <msup> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msup> <mi>ϵ</mi> </mrow> </mfenced> </mrow> </math></EquationSource> </InlineEquation> queries to the block encoding of the Hamiltonian.

An efficient quantum algorithm for ab initio approximations of non-linear response functions

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2025年6月9日

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Tyler Kharazi,
Torin F. Stetina,
Liwen Ko,
Guang Hao Low,
K. Birgitta Whaley

摘要

We develop and analyze a fault-tolerant quantum algorithm for non-linear response properties of molecular and condensed phase systems. We consider a semi-classical description in which the electronic degrees of freedom are treated quantum mechanically and the light is treated as a classical field interacting via the electric dipole approximation. We use the technique of eigenstate filtering, to efficiently resolve excitation energies for dominant dipole transitions. When applied to the electronic structure Hamiltonian with double factorized representation, each significant spectral line can be approximated to a width of ±γ, and to a height of ±ϵ with $O\left({N}^{6}{\eta }^{2}{\gamma }^{-1}{\epsilon }^{-1}\log (1/\epsilon )\right)$ $O (N^{6} η^{2} γ^{- 1} ϵ^{- 1} \log (1 / ϵ))$ queries to the block encoding of the unperturbed electronic structure Hamiltonian for η electrons over N basis functions. These quantities can be used to compute the nth order response functions for non-linear spectroscopies under limited assumptions using $\widetilde{O}\left({N}^{5n+1}{\eta }^{n+1}/{\gamma }^{n}\epsilon \right)$ $\tilde{O} (N^{5 n + 1} η^{n + 1} / γ^{n} ϵ)$ queries to the block encoding of the Hamiltonian.