ORACLE QUANTUM COMPUTING

Building on the work of Deutsch and Jozsa, we construct oracles relati ve to which (1) there is a decision problem that can be solved with ce rtainty in worst-case polynomial time on the quantum computer, yet it cannot be solved classically in probabilistic expected polynomial time if errors are not tolerated, nor even in nondeterministic polynomial time, and (2) there is a decision problem that can be solved in expone ntial time on the quantum computer, which requires double exponential time on all but finitely many instances on any classical deterministic computer.