  The three key elements of a quantum simulation are state preparation, time
evolution, and measurement. While the complexity scaling of time evolution and
measurements are well known, many state preparation methods are strongly
system-dependent and require prior knowledge of the system's eigenvalue
spectrum. Here, we report on a quantum-classical implementation of the
coupled-cluster Green's function (CCGF) method, which replaces explicit ground
state preparation with the task of applying unitary operators to a simple
product state. While our approach is broadly applicable to many models, we
demonstrate it here for the Anderson impurity model (AIM). The method requires
a number of $T$ gates that grows as $ \mathcal{O} \left(N^5 \right)$ per time
step to calculate the impurity Green's function in the time domain, where $N$
is the total number of energy levels in the AIM. Since the number of $T$ gates
is analogous to the computational time complexity of a classical simulation, we
achieve an order of magnitude improvement over a classical CCGF calculation of
the same order, which requires $ \mathcal{O} \left(N^6 \right)$ computational
resources per time step.
