  The parity transformation encodes spin models in the low-energy subspace of a
larger Hilbert-space with constraints on a planar lattice. Applying the Quantum
Approximate Optimization Algorithm (QAOA), the constraints can either be
enforced explicitly, by energy penalties, or implicitly, by restricting the
dynamics to the low-energy subspace via the driver Hamiltonian. While the
explicit approach allows for parallelization with a system-size-independent
circuit depth, the implicit approach shows better QAOA performance. Here we
combine the two approaches in order to improve the QAOA performance while
keeping the circuit parallelizable. In particular, we introduce a modular
parallelization method that partitions the circuit into clusters of subcircuits
with fixed maximal circuit depth, relevant for scaling up to large system
sizes.
