  A universal quantum computer can be used as a simulator capable of predicting
properties of diverse quantum systems. Electronic structure problems in
chemistry offer practical use cases around the hundred-qubit mark. This appears
promising since current quantum processors have reached these sizes. However,
mapping these use cases onto quantum computers yields deep circuits, and for
for pre-fault-tolerant quantum processors, the large number of measurements to
estimate molecular energies leads to prohibitive runtimes. As a result,
realistic chemistry is out of reach of current quantum computers in isolation.
A natural question is whether classical distributed computation can relieve
quantum processors from parsing all but a core, intrinsically quantum component
of a chemistry workflow. Here, we incorporate quantum computations of chemistry
in a quantum-centric supercomputing architecture, using up to 6400 nodes of the
supercomputer Fugaku to assist a Heron superconducting quantum processor. We
simulate the N$_2$ triple bond breaking in a correlation-consistent cc-pVDZ
basis set, and the active-space electronic structure of [2Fe-2S] and [4Fe-4S]
clusters, using 58, 45 and 77 qubits respectively, with quantum circuits of up
to 10570 (3590 2-qubit) quantum gates. We obtain our results using a class of
quantum circuits that approximates molecular eigenstates, and a hybrid
estimator. The estimator processes quantum samples, produces upper bounds to
the ground-state energy and wavefunctions supported on a polynomial number of
states. This guarantees an unconditional quality metric for quantum advantage,
certifiable by classical computers at polynomial cost. For current error rates,
our results show that classical distributed computing coupled to quantum
processors can produce good approximate solutions for practical problems beyond
sizes amenable to exact diagonalization.
