We propose a method for computing space-resolved correlation properties of the two-dimensional Hubbard model within a quantum-classical embedding strategy that uses a noisy, intermediate scale quantum (NISQ) computer to solve the embedded model. While previous approaches were limited to purely local, one-impurity embedded models, requiring at most four qubits and relatively shallow circuits, we solve a two-impurity model requiring eight qubits with an advanced hybrid scheme on top of the variational quantum eigensolver algorithm. This iterative scheme, dubbed natural orbitalization (NOization), gradually transforms the single-particle basis to the approximate natural-orbital basis, in which the ground state can be minimally expressed, at the cost of measuring the one-particle reduced density matrix of the embedded problem. We show that this transformation tends to make the variational optimization of existing (but too deep) Ansatz circuits faster and more accurate, and we propose an Ansatz, the multireference excitation-preserving (MREP) Ansatz, that achieves great expressivity without requiring a prohibitive gate count, thus bridging the gap between hardware-efficient and physically motivated strategies in variational Ansatz design. The one-impurity version of the Ansatz has only one parameter, making the ground-state preparation a trivial step, which supports the optimal character of our approach. Within a rotationally invariant slave boson embedding scheme that requires a minimal number of bath sites and does not require computing the full Green's function, the NOization combined with the MREP Ansatz allow us to compute accurate, space-resolved quasiparticle weights and static self-energies for the Hubbard model even in the presence of noise levels representative of current NISQ processors. This paves the way to a controlled solution of the Hubbard model with larger and larger embedded problems solved by quantum computers.