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Continuous-time quantum walks (CTQWs) on dynamic graphs, referred to as dynamic CTQWs, are a recently introduced universal model of computation that offers a new paradigm in which to envision quantum algorithms. In this work we develop a mapping from dynamic CTQWs to the gate model of computation in the form of an algorithm to convert arbitrary single edge walks and single self loop walks, which are the fundamental building blocks of dynamic CTQWs, to their circuit model counterparts. We use this mapping to introduce an arbitrary quantum state preparation framework based on dynamic CTQWs. Our approach utilizes global information about the target state, relates state preparation to finding the optimal path in a graph, and leads to optimizations in the reduction of controls that are not as obvious in other approaches. Interestingly, classical optimization problems such as the minimal hitting set, minimum spanning tree, and shortest Hamiltonian path problems arise in our framework. We test our methods against uniformly controlled rotations methods, used by Qiskit, and find ours requires fewer CX gates when the target state has a polynomial number of non-zero amplitudes., Comment: Comments are welcome!

The quantum approximate optimization algorithm (QAOA) is a promising quantum algorithm that can be used to approximately solve combinatorial optimization problems. The usual QAOA ansatz consists of an alternating application of the cost and mixer Hamiltonians. In this work, we study how using Hamiltonians other than the usual cost Hamiltonian, dubbed custom phase operators, can affect the performance of QAOA. We derive an expected value formula for QAOA with custom phase operators at p = 1 and show numerically that some of these custom phase operators can achieve higher approximation ratio than the original algorithm implementation. Out of all the graphs tested, 0.036\% of the random custom phase operators, 75.9\% of the subgraph custom phase operators, 95.1\% of the triangle-removed custom phase operators, and 93.9\% of the maximal degree edge-removed custom phase operators have a higher approximation ratio than the original QAOA implementation. This finding opens up the question of whether better phase operators can be designed to further improve the performance of QAOA.

In this paper we consider the scalability of Multi-Angle QAOA with respect to the number of QAOA layers. We found that MA-QAOA is able to significantly reduce the depth of QAOA circuits, by a factor of up to 4 for the considered data sets. However, MA-QAOA is not optimal for minimization of the total QPU time. Different optimization initialization strategies are considered and compared for both QAOA and MA-QAOA. Among them, a new initialization strategy is suggested for MA-QAOA that is able to consistently and significantly outperform random initialization used in the previous studies.