Efficient Sparse State Preparation via Quantum Walks

Abstract: 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 efficient sparse 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 the state-of-the-art ancilla free sparse state preparation method and a uniformly controlled rotation method, which is used by Qiskit, and find ours requires fewer CX gates when the target state has a polynomial number of non-zero amplitudes.