  We present calculations of the ground and excited state energies of spin
defects in solids carried out on a quantum computer, using a hybrid
classical/quantum protocol. We focus on the negatively charged nitrogen vacancy
center in diamond and on the double vacancy in 4H-SiC, which are of interest
for the realization of quantum technologies. We employ a recently developed
first-principle quantum embedding theory to describe point defects embedded in
a periodic crystal, and to derive an effective Hamiltonian, which is then
transformed to a qubit Hamiltonian by means of a parity transformation. We use
the variational quantum eigensolver (VQE) and quantum subspace expansion
methods to obtain the ground and excited states of spin qubits, respectively,
and we propose a promising strategy for noise mitigation. We show that by
combining zero-noise extrapolation techniques and constraints on electron
occupation to overcome the unphysical state problem of the VQE algorithm, one
can obtain reasonably accurate results on near-term-noisy architectures for
ground and excited state properties of spin defects.
