A variational quantum eigensolver for dynamic correlation functions

Abstract

Recent practical approaches for the use of current generation noisy quantum devices in the simulation of quantum many-body problems have been dominated by the use of a variational quantum eigensolver (VQE). These coupled quantum-classical algorithms leverage the ability to perform many repeated measurements to avoid the currently prohibitive gate depths often required for exact quantum algorithms, with the restriction of a parameterized circuit to describe the states of interest. In this work, we show how the calculation of zero-temperature dynamic correlation functions defining the linear response characteristics of quantum systems can also be recast into a modified VQE algorithm, which can be incorporated into the current variational quantum infrastructure. This allows for these important physical expectation values describing the dynamics of the system to be directly converged on the frequency axis, and they approach exactness over all frequencies as the flexibility of the parameterization increases. The frequency resolution hence does not explicitly scale with gate depth, which is approximately twice as deep as a ground state VQE. We apply the method to compute the single-particle Green's function of ab initio dihydrogen and lithium hydride molecules, and demonstrate the use of a practical active space embedding approach to extend to larger systems. While currently limited by the fidelity of two-qubit gates, whose number is increased compared to the ground state algorithm on current devices, we believe the approach shows potential for the extraction of frequency dynamics of correlated systems on near-term quantum processors.

Hybrid quantum-classical algorithm for computing imaginary-time correlation functions

“…Recently, a variational quantum algorithm to directly obtain Green's function in the frequency domain was developed [16]. This algorithm may be more general in that it directly computes the real-time Green's function as well as the imaginary-frequency Green's function.…”

“…Equation ( 18) can be measured using the quantum circuit in Fig. 3 [16,27], which requires one ancilla qubit. Let p 0 /p 1 be the probability of measuring 0/1 in the ancilla qubit.…”

“…However, these proposals assume a fault-tolerant quantum computer, which is beyond the capabilities of quantum devices with limited hardware resources such as noisy intermediate-scale quantum (NISQ) devices [14]. Therefore, the development of methods to calculate Green's function using quantum devices with limited hardware resources has been actively pursued [15,16].…”

“…This is because the imaginary-time formalism allows us to discretize the environment with fewer auxiliary bath sites [17][18][19] and hence qubits. Although a recently proposed algorithm has been used to compute the imaginary-frequency Green's function [16], it requires the evaluation of the expectation value of the square of the Hamiltonian, which may be expensive for a large impurity model. Thus, efficient methods for computing the imaginary-time Green's function using quantum devices with limited hardware resources still remain to be explored.…”

Quantum Algorithms for Ground-State Preparation and Green's Function Calculation

“…For example, the spectral density and local density of states (LDOS) of the system is given by the imaginary part of the retarded single-particle Green's function Im G R (ω). Various methods have been recently proposed for calculating Green's functions and other related quantities [20][21][22][23][24][25] using quantum computing devices. For example, the Gaussian integral transformation (GIT) [21] is used to compute the spectral density of the system by employing the qubitization technique, while statistical sampling and QPE is recently used to compute the single-particle and two-particle Green's functions of simple molecules [22,23] and similar approach is applied to compute other dynamic linear response functions [20].…”