Approximating dynamical correlation functions with constant depth quantum circuits

Abstract

One of the most important quantities characterizing the microscopic properties of quantum systems are dynamical correlation functions. These correlations are obtained by time-evolving a perturbation of an eigenstate of the system, typically the ground state. In this work, we study approximations of these correlation functions that do not require time dynamics. We show that having access to a circuit that prepares an eigenstate of the Hamiltonian, it is possible to approximate the dynamical correlation functions up to exponential accuracy in the complex frequency domain $\omega =\Re (\omega )+i\Im (\omega )$, on a strip above the real line $\Im (\omega )=0$. We achieve this by exploiting the continued fraction representation of the dynamical correlation functions as functions of frequency $\omega$, where the level $k$ approximant can be obtained by measuring a weight $O(k)$ operator on the eigenstate of interest. In the complex $\omega$ plane, we show how this approach allows to determine approximations to correlation functions with accuracy that increases exponentially with $k$. We analyse two algorithms to generate the continuous fraction representation in scalar or matrix form, starting from either one or many initial operators. We prove that these algorithms generate an exponentially accurate approximation of the dynamical correlation functions on a region sufficiently far away from the real frequency axis. We present numerical evidence of these theoretical results through simulations of small lattice systems. We comment on the stability of these algorithms with respect to sampling noise in the context of quantum simulation using quantum computers.

31 pages, 10 figures

References in corpus (23)

• Electronic Structure Calculations with Dynamical Mean-Field Theory: A Spectral Density Functional Approach
• Predicting Many Properties of a Quantum System from Very Few Measurements
• Nonequilibrium dynamical mean-field theory and its applications
• A Universal Operator Growth Hypothesis
• Hybrid quantum-classical approach to correlated materials
• Efficient estimation of Pauli observables by derandomization
• Fermionic partial tomography via classical shadows
• Observing ground-state properties of the Fermi-Hubbard model using a scalable algorithm on a quantum computer
• The Quantum Wasserstein Distance of Order 1
• Calculation of the Green's function on near-term quantum computers
• A theory of quantum subspace diagonalization
• Quantum Simulations of Physics Problems
• Non-linear quantum-classical scheme to simulate non-equilibrium strongly correlated fermionic many-body dynamics
• Construction of Green's functions on a quantum computer: applications to molecular systems
• An Algorithm for Quantum Computation of Particle Decays
• Effective calculation of the Green's function in the time domain on near-term quantum processors
• Hybrid quantum-classical algorithm for computing imaginary-time correlation functions
• Improving the accuracy and efficiency of quantum connected moments expansions
• Analysis of quantum Krylov algorithms with errors
• Orthogonal polynomials, a short introduction
• Sampling Error Analysis in Quantum Krylov Subspace Diagonalization
• Quantum Computed Green's Functions using a Cumulant Expansion of the Lanczos Method
• Super-resolution of Green's functions on noisy quantum computers

Cited by in corpus (1)

• A circuit-differentiation framework for Green's functions on quantum computers