The Lie algebra of XY-mixer topologies and warm starting QAOA for constrained optimization

The XY-mixer is widely used in quantum computing, particularly in variational quantum algorithms like the Quantum Alternating Operator Ansatz (QAOA). It is effective for solving Cardinality Constrained Optimization problems, a broad class of NP-hard tasks. We provide explicit decompositions of the dynamical Lie algebras (DLAs) for various XY-mixer topologies. When these DLAs decompose into polynomial-sized Lie algebras, they are efficiently trainable, such as in cycle XY-mixers with arbitrary RZ. In contrast, all-to-all XY-mixers or the inclusion of arbitrary RZZ gates lead to exponentially large DLAs, making training intractable. We warm-start optimization by pre-training on smaller, polynomially sized DLAs via gate-generator restriction, improving convergence and yielding better solution quality for both shared-angle and multi-angle QAOA instances. Multi-angle QAOA is also shown to be more performant than shared-angle QAOA on the instances considered.