Simulating Quantum Walk Hamiltonians without Pauli Decomposition

In this work, we present a new algorithm for generating quantum circuits that efficiently implement continuous time quantum walks on arbitrary simple sparse graphs. The algorithm, called matching decomposition, works by decomposing a continuous-time quantum walk Hamiltonian into a collection of exactly implementable Hamiltonians corresponding to matchings in the underlying graph followed by a novel graph compression algorithm that merges edges in the graph. Lastly, we convert the walks to a circuit and Trotterize over these components. The dynamics of the walker on each edge in the matching can be implemented in the circuit model as sequences of CX and CRx gates. We do not use Pauli decomposition when implementing walks along each matching. Furthermore, we compare matching decomposition to a standard Pauli-based simulation pipeline and find that matching decomposition consistently yields substantial resource reductions, requiring up to 43% fewer controlled gates and up to 54% shallower circuits than Pauli decomposition across multiple graph families. Finally, we also present examples and theoretical results for when matching decomposition can exactly simulate a continuous-time quantum walk on a graph.