This paper presents a quantum-centric supercomputing approach for simulating chemistry on a large-scale quantum processor. The authors use a 6400-node supercomputer, Fugaku, to assist a quantum processor with a Heron superconducting processor in simulating chemical systems. They simulate the breaking of the N₂ triple bond 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. The results are obtained using a class of quantum circuits that approximate molecular eigenstates and a hybrid estimator that processes quantum samples to produce upper bounds to the ground-state energy and wavefunctions. This guarantees an unconditional quality metric for quantum advantage, certifiable by classical computers at polynomial cost. The authors propose a self-consistent configuration recovery technique to improve the signal-to-noise ratio in quantum measurements. This technique allows for the probabilistic partial recovery of noiseless configuration samples from noisy quantum measurements. The configuration recovery scheme is inspired by the structure of chemistry problems and is used to improve the accuracy of quantum simulations. The method is applied to simulate the dissociation of N₂ and the electronic structure of [2Fe-2S] and [4Fe-4S] clusters. The results show that classical distributed computing combined with quantum computers can produce good approximate solutions for practical problems beyond sizes amenable to exact diagonalization. The authors also discuss the advantages and limitations of their method, including its ability to handle large-scale quantum simulations and its potential for precision many-body physics. They compare their results with classical methods such as Heat-Bath Configuration Interaction (HCI) and show that their method achieves similar accuracy within tens of mE_h. The method is also applied to simulate the electronic structure of iron-sulfur clusters, which are important cofactors in biological processes. The results show that the method can accurately simulate the electronic structure of these clusters, even in the presence of strong static correlation. The authors conclude that their method provides a promising approach for simulating chemistry on quantum computers, combining the strengths of quantum and classical computing to achieve accurate and efficient simulations of complex chemical systems. The method is particularly useful for systems that are beyond the reach of exact diagonalization and can benefit from the scalability of quantum computing. The results demonstrate the potential of quantum computing for solving complex chemical problems and highlight the importance of combining quantum and classical computing for achieving practical quantum advantage.This paper presents a quantum-centric supercomputing approach for simulating chemistry on a large-scale quantum processor. The authors use a 6400-node supercomputer, Fugaku, to assist a quantum processor with a Heron superconducting processor in simulating chemical systems. They simulate the breaking of the N₂ triple bond 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. The results are obtained using a class of quantum circuits that approximate molecular eigenstates and a hybrid estimator that processes quantum samples to produce upper bounds to the ground-state energy and wavefunctions. This guarantees an unconditional quality metric for quantum advantage, certifiable by classical computers at polynomial cost. The authors propose a self-consistent configuration recovery technique to improve the signal-to-noise ratio in quantum measurements. This technique allows for the probabilistic partial recovery of noiseless configuration samples from noisy quantum measurements. The configuration recovery scheme is inspired by the structure of chemistry problems and is used to improve the accuracy of quantum simulations. The method is applied to simulate the dissociation of N₂ and the electronic structure of [2Fe-2S] and [4Fe-4S] clusters. The results show that classical distributed computing combined with quantum computers can produce good approximate solutions for practical problems beyond sizes amenable to exact diagonalization. The authors also discuss the advantages and limitations of their method, including its ability to handle large-scale quantum simulations and its potential for precision many-body physics. They compare their results with classical methods such as Heat-Bath Configuration Interaction (HCI) and show that their method achieves similar accuracy within tens of mE_h. The method is also applied to simulate the electronic structure of iron-sulfur clusters, which are important cofactors in biological processes. The results show that the method can accurately simulate the electronic structure of these clusters, even in the presence of strong static correlation. The authors conclude that their method provides a promising approach for simulating chemistry on quantum computers, combining the strengths of quantum and classical computing to achieve accurate and efficient simulations of complex chemical systems. The method is particularly useful for systems that are beyond the reach of exact diagonalization and can benefit from the scalability of quantum computing. The results demonstrate the potential of quantum computing for solving complex chemical problems and highlight the importance of combining quantum and classical computing for achieving practical quantum advantage.