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arXiv:1804.01505 · Physical Review C

Dynamic linear response quantum algorithm

Alessandro Roggero, Joseph Carlson

Published Sep 13, 2019• 103 citations
arXivHTMLPDF

Abstract

The dynamic linear response of a quantum system is critical for understanding both the structure and dynamics of strongly interacting quantum systems, including neutron scattering from materials, photon and electron scattering from atomic systems, and electron and neutrino scattering by nuclei. We present a general algorithm for universal quantum computers to calculate the dynamic linear response function with controlled errors and to obtain information about specific final states that can be directly compared to experimental observations.

DOI
10.1103/physrevc.100.034610

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The dynamic linear response of a quantum system is critical for understanding both the structure and dynamics of strongly interacting quantum systems, including neutron scattering from materials, photon and electron scattering from atomic systems, and electron and neutrino scattering by nuclei. We present a general algorithm for universal quantum computers to calculate the dynamic linear response function with controlled errors and to obtain information about specific final states that can be directly compared to experimental observations.
We present a quantum algorithm to obtain the response of the atomic nucleus to a small external electromagnetic perturbation. The Hamiltonian of the system is presented by a harmonic oscillator, and the linear combination of unitaries (LCU) based method is utilized to simulate the Hamiltonian on the quantum computer. The output of the Hamiltonian simulation is utilized in calculating the dipole response with the SWAP test algorithm. The results of the response function computed using the quantum algorithm are compared with the experimental data and provide a good agreement. We show the results for 120^{120}120Sn and 208^{208}208Pb to corroborate with the experimental data in Sn and Pb region and also compare the results with those obtained using the conventional linear response theory.
The description of quantum many-body dynamics is extremely challenging on classical computers, as it can involve many degrees of freedom. On the other hand, the time evolution of quantum states is a natural application for quantum computers that are designed to efficiently perform unitary transformations. In this paper, we study quantum algorithms for response functions, relevant for describing different reactions governed by linear response. We focus on nuclear-physics applications and consider a qubit-efficient mapping on the lattice, which can efficiently represent the large volumes required for realistic scattering simulations. For the case of a contact interaction, we develop an algorithm for time evolution based on the Trotter approximation that scales logarithmically with the lattice size, and is combined with quantum phase estimation. We eventually focus on the nuclear two-body system and a typical response function relevant for electron scattering as an example. We also investigate ground-state preparation and examine the total circuit depth required for a realistic calculation and the hardware noise level required to interpret the signal.
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Response functions are a fundamental aspect of physics; they represent the link between experimental observations and the underlying quantum many-body state. However, this link is often under-appreciated, as the Lehmann formalism for obtaining response functions in linear response has no direct link to experiment. Within the context of quantum computing, and via a linear response framework, we restore this link by making the experiment an inextricable part of the quantum simulation. This method can be frequency- and momentum-selective, avoids limitations on operators that can be directly measured, and can be more efficient than competing methods. As prototypical examples of response functions, we demonstrate that both bosonic and fermionic Green's functions can be obtained, and apply these ideas to the study of a charge-density-wave material on the ibm_auckland superconducting quantum computer. The linear response method provides a robust framework for using quantum computers to study systems in physics and chemistry.
Response functions are a fundamental aspect of physics; they represent the link between experimental observations and the underlying quantum many-body state. However, this link is often under-appreciated, as the Lehmann formalism for obtaining response functions in linear response has no direct link to experiment. Within the context of quantum computing, and by using a linear response framework, we restore this link by making the experiment an inextricable part of the quantum simulation. This method can be frequency- and momentum-selective, avoids limitations on operators that can be directly measured, and is ancilla-free. As prototypical examples of response functions, we demonstrate that both bosonic and fermionic Green's functions can be obtained, and apply these ideas to the study of a charge-density-wave material on ibm_auckland. The linear response method provides a robust framework for using quantum computers to study systems in physics and chemistry. It also provides new paradigms for computing response functions on classical computers.
Introduction There are countless situations in physics when one is interested in calculating the response of a system to a small time-dependent perturbation acting on it. With some luck the response can be expanded in a power series of the strength of the perturbation, so that, to first order, it is a linear function of the latter. To compute this function is the objective of the linear response theory (LRT). Linear response theory has many important applications to the study of electronic matter. Virtually all interactions of electrons with experimental probes (electromagnetic fields, beams of particles) can be regarded as small perturbations to the system: if they were not, one would not be probing the system, but the system modified by the probe. Consequently, the results of these experiments can be expressed in terms of linear response functions, which are properties of the unperturbed system. In particular it will turn out that the analytic structure of these functions is entirely determined by the eigenvalues and eigenfunctions of the unperturbed system. Conversely, a measure of the linear response as a function of the frequency of the perturbation enables us to determine the excitation energies of the system. Beside being a cornerstone for the theory of single-particle properties to be developed in Chapter 8, the linear response functions also provide invaluable information in their own right. For example, as we will discuss in Chapter 5, the extent to which an external electrostatic potential is reduced by screening is controlled by the dynamical dielectric function which, in turn, is determined by the density–density response function.