An Initialization-free Quantum Algorithm for General Abelian Hidden Subgroup Problem

Hidden Subgroup Problem(HSP) seeks to identify an unknown subgroup H of a group G for a given injective function f defined on cosets of H. Here we present an initialization-free quantum algorithm for solving HSP in the case where G is a finite abelian group. Our algorithm can adopt an arbitrary unknown mixed state as the auxiliary register and removes the need for initialization while preserving computational cost comparable to existing methods. Our algorithm also restores the state of the auxiliary register to its original form after completing the computations. Since the recovered state can be utilized for other operations, a single preparation of the auxiliary register in an arbitrarily unknown mixed state is sufficient to execute the iterative procedure in solving hidden subgroup problems. This approach provides a promising direction for improving quantum algorithm efficiency by reducing operational time of initialization.