  We introduce parity quantum optimization with the aim of solving optimization
problems consisting of arbitrary $k$-body interactions and side conditions
using planar quantum chip architectures. The method introduces a decomposition
of the problem graph with arbitrary $k$-body terms using generalized closed
cycles of a hypergraph. Side conditions of the optimization problem in form of
hard constraints can be included as open cycles containing the terms involved
in the side conditions. The generalized parity mapping thus circumvents the
need to translate optimization problems to a quadratic unconstrained binary
optimization problem (QUBO) and allows for the direct encoding of higher-order
constrained binary optimization problems (HCBO) on a square lattice and full
parallelizability of gates.
