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