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We propose a divide-and-conquer method for the quantum-classical hybridalgorithm to solve larger problems with small-scale quantum computers.Specifically, we concatenate a variational quantum eigensolver (VQE) with areduction in the system dimension, where the interactions between dividedsubsystems are taken as an effective Hamiltonian expanded by the reduced basis.Then the effective Hamiltonian is further solved by VQE, which we call deepVQE. Deep VQE allows us to apply quantum-classical hybrid algorithms onsmall-scale quantum computers to large systems with strong intra-subsysteminteractions and weak inter-subsystem interactions, or strongly correlated spinmodels on large regular lattices. As proof-of-principle numericaldemonstrations, we use the proposed method for quasi one-dimensional models,including one-dimensionally coupled 12-qubit Heisenberg anti-ferromagneticmodels on Kagome lattices as well as two-dimensional Heisenberganti-ferromagnetic models on square lattices. The largest problem size of 64qubits is solved by simulating 20-qubit quantum computers with a reasonablygood accuracy ~ a few %. The proposed scheme enables us to handle the problemsof >1000 qubits by concatenating VQEs with a few tens of qubits. While it isunclear how accurate ground state energy can be obtained for such a largesystem, our numerical results on a 64-qubit system suggest that deep VQEprovides a good approximation (discrepancy within a few percent) and has a roomfor further improvement. Therefore, deep VQE provides us a promising pathway tosolve practically important problems on noisy intermediate-scale quantumcomputers.

Deep Variational Quantum Eigensolver: a divide-and-conquer method for solving a larger problem with smaller size quantum computers