However, FQE might not be suitable for advantageous quantum chemistry demonstrations due to the limitations of NISQ devices. Full quantum eigensolver (FQE), in which the complexity of basic gate operators is polylogarithmic in the number of spin-orbitals, represents another fully quantum algorithm to simulate molecular systems 10. However, QPE requires millions of qubits and quantum gates even for relatively small systems, making the algorithm unsuitable for practical applications on near-term noisy intermediate-scale quantum (NISQ) devices 9. For example, the quantum phase estimation (QPE) algorithm 7 represents the natural translation of the full configuration interaction (FCI) procedure to quantum computers 8. The direct solution to these problems on currently available quantum computer hardware is intractable. One of the most promising and immediate applications of quantum computers is solving classically intractable quantum chemistry problems 5, 6. The reason is that the solution space (i.e., the Fock space) grows factorially with the system size (e.g., the number of electrons and basis functions). Even though modern quantum many-body methods, such as density matrix renormalization group (DMRG) 2, selected configuration interaction (sCI) 3, and coupled-cluster (CC) theory 4 methods can solve larger systems, their applications are still limited to dozens of electrons and they are subject to truncation or approximations. However, since the invention of classical digital computers in the early 1940s, the exact numerical solution of this central quantum mechanical equation remains infeasible for systems roughly having more than 12 electrons distributed on 184 spin-orbitals 1. Such eigenvalues and eigenvectors can be obtained from the solution of the time-independent electronic Schrödinger equation. One of the major goals of computational chemistry is the development of methods and algorithms for the calculation of the molecular electronic ground and excited state energies and corresponding wave functions from first-principles. Proof-of-principle demonstrations are presented for several molecular systems based on quantum simulators as well as IBM quantum devices. ClusterVQE therefore allows exact simulation of the problem by using fewer qubits and shallower circuit depth at the cost of additional classical resources, making it a potential leader for quantum chemistry simulations on NISQ devices. The clusters are obtained based on mutual information reflecting maximal entanglement between qubits, whereas inter-cluster correlation is taken into account via a new “dressed” Hamiltonian. Our ClusterVQE algorithm splits the initial qubit space into clusters which are further distributed on individual (shallower) quantum circuits. Here we present an approach to reduce quantum circuit complexity in VQE for electronic structure calculations. The practical realization is limited by the complexity of quantum circuits. The variational quantum eigensolver (VQE) is one of the most promising algorithms to find eigenstates of a given Hamiltonian on noisy intermediate-scale quantum devices (NISQ).
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