  The iterative qubit coupled cluster (iQCC) method is a systematic variational
approach to solve the electronic structure problem on universal quantum
computers. It is able to use arbitrarily shallow quantum circuits at expense of
iterative canonical transformation of the Hamiltonian and rebuilding a circuit.
Here we present a variety of a posteriori corrections to the iQCC energies to
reduce the number of iterations to achieve the desired accuracy. Our energy
corrections are based on a low-order perturbation theory series that can be
efficiently evaluated on a classical computer. Moreover, capturing a part of
the total energy perturbatively, allows us to formulate the qubit active-space
concept, in which only a subset of all qubits is treated variationally. As a
result, further reduction of quantum resource requirements is achieved. We
demonstrate the utility and efficiency of our approach numerically on the
examples of 10-qubit N$_2$ molecule dissociation, the 24-qubit H$_2$O symmetric
stretch, and 56-qubit singlet-triplet gap calculations for the technologically
important complex, tris-(2-phenylpyridine)iridium(III), Ir(ppy)$_3$.
