  Quantum spin systems may offer the first opportunities for beyond-classical
quantum computations of scientific interest. While general quantum simulation
algorithms likely require error-corrected qubits, there may be applications of
scientific interest prior to the practical implementation of quantum error
correction. The variational quantum eigensolver (VQE) is a promising approach
to finding energy eigenvalues on noisy quantum computers. Lattice models are of
broad interest for use on near-term quantum hardware due to the sparsity of the
number of Hamiltonian terms and the possibility of matching the lattice
geometry to the hardware geometry. Here, we consider the Kitaev spin model on a
hardware-native square-octagon qubit connectivity map, and examine the
possibility of efficiently probing its rich phase diagram with VQE approaches.
By benchmarking different choices of variational Ansatz states and classical
optimizers, we illustrate the advantage of a mixed optimization approach using
the Hamiltonian variational Ansatz (HVA) and the potential of probing the
system's phase diagram using VQE. We further demonstrate the implementation of
HVA circuits on Rigetti's Aspen-9 chip with error mitigation.
