  We introduce a hybrid quantum-classical variational algorithm to simulate
ground-state phase diagrams of frustrated quantum spin models in the
thermodynamic limit. The method is based on a cluster-Gutzwiller ansatz where
the wave function of the cluster is provided by a parameterized quantum circuit
whose key ingredient is a two-qubit real XY gate allowing to efficiently
generate valence-bonds on nearest-neighbor qubits. Additional tunable
single-qubit Z- and two-qubit ZZ-rotation gates allow the description of
magnetically ordered and paramagnetic phases while restricting the variational
optimization to the U(1) subspace. We benchmark the method against the J1-J2
Heisenberg model on the square lattice and uncover its phase diagram, which
hosts long-range ordered Neel and columnar anti-ferromagnetic phases, as well
as an intermediate valence-bond solid phase characterized by a periodic pattern
of 2x2 strongly-correlated plaquettes. Our results show that the convergence of
the algorithm is guided by the onset of long-range order, opening a promising
route to synthetically realize frustrated quantum magnets and their quantum
phase transition to paramagnetic valence-bond solids with currently developed
superconducting circuit devices.
