The Lie Algebra of XY-mixer Topologies and Warm Starting QAOA for Constrained Optimization

The XY-mixer has widespread utilization in modern quantum computing, including in variational quantum algorithms, such as Quantum Alternating Operator Ansatz (QAOA). The XY ansatz is particularly useful for solving Cardinality Constrained Optimization tasks, a large class of important NP-hard problems. First, we give explicit decompositions of the dynamical Lie algebras (DLAs) associated with a variety of $XY$-mixer topologies. When these DLAs admit simple Lie algebra decompositions, they are efficiently trainable. An example of this scenario is a ring $XY$-mixer with arbitrary $R_Z$ gates. Conversely, when we allow for all-to-all $XY$-mixers or include $R_{ZZ}$ gates, the DLAs grow exponentially and are no longer efficiently trainable. We provide numerical simulations showcasing these concepts on Portfolio Optimization, Sparsest $k$-Subgraph, and Graph Partitioning problems. These problems correspond to exponentially-large DLAs and we are able to warm-start these optimizations by pre-training on polynomial-sized DLAs by restricting the gate generators. This results in improved convergence to high quality optima of the original task, providing dramatic performance benefits in terms of solution sampling and approximation ratio on optimization tasks for both shared angle and multi-angle QAOA.