  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.
