Block Encoding of Sparse Matrices via Coherent Permutation

Abstract: Block encoding of sparse matrices underpins powerful quantum algorithms such as quantum singular value transformation, Hamiltonian simulation, and quantum linear solvers, but its efficient gate-level implementation for arbitrary sparse matrices remains a major challenge. We introduce a unified framework that overcomes the key obstacles of multi-controlled X gates overhead, amplitude reordering, and hardware connectivity, enabling efficient block encoding for arbitrary sparse matrices with explicit gate-level constructions. Central to our approach are a novel connection with combinatorial optimization, which enables systematic assignment of control qubits to achieve nearest-neighbor connectivity, and coherent permutation operators that preserve superposition while enabling amplitude reordering. We demonstrate our methods on structured sparse matrices, showing significant reductions in circuit depth and control overhead, thereby bridging the gap between theoretical formulations and practical circuit implementations for quantum algorithms.