  Quantum computing promises a computational advantage over classical methods
in electronic-structure calculations, with expected applications in drug design
and materials science. Accessing a quantitative description of chemical systems
while minimizing quantum resources, such as the number of qubits, is an
essential challenge given the limited capabilities of current quantum
processors. We provide a shortcut towards quantum computations at chemical
accuracy by approaching the complete-basis-set limit (CBS) through integrating
density-functional theory into quantum algorithms via density-based basis-set
corrections coupled to basis-sets crafted on-the-fly and specifically adapted
to a given system/user-defined qubit budget. The approach self-consistently
accelerates the basis-set convergence, improving electronic densities,
ground-state energies, and first-order properties such as dipole moments. It
can also serve as a classical, a posteriori, energy correction to quantum
hardware calculations. The strategy is assessed using GPU-accelerated
state-vector emulation up to 32 qubits. We converge the ground-state energies
of four systems (He, Be, H$_2$, LiH) within chemical accuracy of the CBS
full-configuration-interaction reference, while offering a systematic increase
of accuracy beyond a double-zeta quality for various molecules up to the H$_8$
hydrogen chain. We also obtain dissociation curves for H$_2$ and LiH that reach
the CBS limit whereas for the challenging simulation of the N$_2$ triple-bond
breaking, we achieve a near-triple-zeta quality at the cost of a minimal
basis-set. This hybrid strategy allows us to obtain quantitative results that
would otherwise require brute-force quantum simulations using far more than 100
logical qubits, thereby opening up opportunities to explore real-world
chemistry with reasonable computational resources.
