The multi-angle quantum approximate optimization algorithm (ma-QAOA) is a recently introduced algorithm that gives at least the same approximation ratio as the quantum approximate optimization algorithm (QAOA) and, in most cases, gives a significantly higher approximation ratio than QAOA. One drawback to ma-QAOA is that it uses significantly more classical parameters than QAOA, so the classical optimization component more complex. In this paper, we motivate a new parameter initialization strategy in which angles are initially randomly set to multiples of π/8 between −π and π and this vector is used to seed one round of BFGS. We find that this parameter initialization strategy gives average approximation ratios of 0.900, 0.982, and 0.997 for p=1,2,3 layers of ma-QAOA. This is comparable to the average approximation ratios of ma-QAOA where the optimal parameters are found using BFGS with 1 random starting seed, which are 0.900, 0.982, and 0.996. We also test another parameter initialization strategy in which angles corresponding to maximal degree vertices in the graph are set to 0 while all other are randomly initialized to random multiples of π/8. Using this strategy, the average approximation ratios are 0.897, 0.984, and 0.997.
The standard quantum approximate optimization algorithm (QAOA) has a limited expressive power in shallow circuits, whereas naively increasing the number of layers leads to greater quantum circuit depth and higher noise sensitivity. To address this trade-off, based on the ideas of the multi-angle quantum approximate optimization algorithm (MA-QAOA) and the RY-layer-assisted quantum approximate optimization algorithm (RY-QAOA), this paper proposes an improved algorithm, i.e., MA-RY-QAOA, which integrates parameter multangularization and RY-assisted architecture. In MA-RY-QAOA, the problem block incorporates edge parameter multangularization together with RY extension, while the mixer block implements qubit parameter multangularization. For layer number p=1 and node size n=4~12 , we conduct 50 simulation trials on maximum cut problem respectively for random graph with edge probability of 0.8, complete graph, 2-regular graph and 4-regular graph, by using the average approximation ratio as the evaluation metric. The simulation trial results show that, across these four class graphs, the average approximation ratio of MA-RY-QAOA and MA-QAOA significantly outperforms standard QAOA, RY-QAOA, and QAOA+ for overwhelming majority of node sizes; with respect to complete graph, the average approximation ratio of MA-QAOA outperforms that of MA-RY-QAOA for all node sizes; with regard to 2-regular graph, the average approximation ratio of MA-RY-QAOA exceeds that of MA-QAOA for all node sizes; with regard to 4-regular graph, the average approximation ratio of MA-RY-QAOA takes advantage over that of MA-QAOA for overwhelming majority of node sizes; concerning random graph, the average approximation ratio of MA-RY-QAOA is superior to that of MA-QAOA for both small node sizes and large node sizes, but is inferior to that of MA-QAOA for medium node sizes. Therefore, MA-RY-QAOA provides a useful reference for the structure design of shallow-layer QAOA on noisy intermediate-scale quantum (NISQ) devices.
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.