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arXiv:2004.04889 · Physical Review A

Spectral-density estimation with the Gaussian integral transform

Alessandro Roggero

Published Aug 14, 2020• 41 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.102.022409

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The spectral-density operator \ensuremathρ\ifmmode^\elseˆ\fi(\ensuremathω)=\ensuremathδ(\ensuremathω\ensuremath−H\ifmmode^\elseˆ\fi)\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}(\ensuremath{\omega})=\ensuremath{\delta}(\ensuremath{\omega}\ensuremath{-}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H})\ensuremathρ\ifmmode^\elseˆ\fi​(\ensuremathω)=\ensuremathδ(\ensuremathω\ensuremath−H\ifmmode^\elseˆ\fi​) 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 \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ and error \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ using O(log2(1/\ensuremathϵ)[log2(1/\ensuremathΔ)+log2(1/\ensuremathϵ)]/\ensuremathΔ)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)O(log2​(1/\ensuremathϵ)[log2​(1/\ensuremathΔ)+log2​(1/\ensuremathϵ)]​/\ensuremathΔ) 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.
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.
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.
Modern quantum algorithms are increasingly formulated as coherent procedures for implementing polynomial transformations of operators and singular values. This perspective provides a powerful and unifying language for quantum algorithm design, connecting a wide range of distinct problems through five closely related key tools: block-encoding, qubitization, QSP, QSVT and GQSP. Block-encoding embeds non-unitary matrices into larger unitaries; qubitization converts block-encodings into structured operators; QSP, QSVT and GQSP enable polynomial transformations with near-optimal query complexity. Together, these techniques form a general toolkit for transforming matrix functions into implementable quantum circuits. This paper develops these techniques from first principles as a unified framework for constructing quantum algorithms. We apply this framework to representative applications to highlight design principles and demonstrate how distinct algorithms can be constructed from a unified sequence of operator transformations. A central contribution is a systematic decision workflow for selecting the appropriate approach according to the operator structure and the desired transformation polynomial. This perspective clarifies when direct GQSP or through qubitization, or Laurent expansion, or QSVT is most appropriate. We organize algorithmic design into an end-to-end pipeline: identifying the target matrix function, constructing an appropriate block-encoding, determining the relevant spectral domain, designing a polynomial or Laurent-polynomial approximation, synthesizing the phase factors, and translating the transformation into an executable quantum circuit. By applying this unified framework to example applications, we showcase a practical methodology for reasoning, designing, and implementing quantum algorithms based on polynomial transformations.
Neutrino-nucleus cross section uncertainties are expected to be a dominant systematic in future accelerator neutrino experiments. The cross sections are determined by the linear response of the nucleus to the weak interactions of the neutrino, and are dominated by energy and distance scales of the order of the separation between nucleons in the nucleus. These response functions are potentially an important early physics application of quantum computers. Here we present an analysis of the resources required and their expected scaling for scattering cross section calculations. The current estimates of Trotter steps needed to achieve an energy resolution of 10 MeV and the number of CNOT gates for analyzing 40Ar^{40}\mathrm{Ar}40Ar highlights the need for significant improvements in algorithms. We also examine simple small-scale neutrino-nucleus models on modern quantum hardware. In this paper, we use variational methods to obtain the ground state of a three nucleon system (the triton) and then implement the relevant time evolution. To tame the errors in present-day NISQ devices, we explore the use of different error-mitigation techniques to increase the fidelity of the calculations.