Cheap partial diffusers for arbitrary initial states

Extend the recursive quantum-search construction to arbitrary initial quantum states while retaining partial diffusers that are cheaper to implement than a global reflection.

Background

The construction assumes that the initial state factorises over the chosen product partition, which allows partial diffusers to act on progressively smaller subspaces. An arbitrary initial state can be transformed into product form using a disentangling unitary, but the corresponding partial diffusers must then be conjugated by that unitary at every level. Such conjugation can make each diffuser as costly as the global reflection and can dominate the non-oracle complexity. The paper therefore identifies extending the method to arbitrary initial states without losing the implementation advantage of partial diffusers as an unresolved problem.