Phase Information in Quantum Oracle Computing

Computational devices may be supplied with external sources of information (oracles). Quantum oracles may transmit phase information which is available to a quantum computer but not a classical computer. One consequence of this observation is that there is an oracle which is of no assistance to a classical computer but which allows a quantum computer to solve undecidable problems. Thus useful relativized separations between quantum and classical complexity classes must exclude the transmission of phase information from oracle to computer. Deutsch [1] introduced the notion of a universal quantum computer. This machine is the natural generalization of a (reversible) Turing machine to the quantum world. The universal quantum computer differs from its classical counterpart in that it may evolve in a superposition of states. Thus there is the prospect that a single quantum computer may be able to carry out many simultaneous calculations. Unfortunately, the naive idea of quantum parallelism is defeated by the need to make a measurement at the end of the computation so that only one of the many parallel computations is available to the (classical) user. Nonetheless, a more restricted notion of quantum parallelism was recently introduced by Deutsch and Jozsa [2]. They show that relative to an ‘oracle’, a quantum computer may solve certain problems much faster than a Turing machine equipped with the same oracle. An oracle is an auxiliary device to which the computer can address queries and receive YES or NO answers. Deutsch and Jozsa envisioned a quantum oracle which may receive a superposition of questions and return a superposition of answers. Later Berthiaume and Brassard [3, 4] refined the notion of ‘oracle quantum computing’ and obtained results separating conventional (classical) and quantum complexity classes relative to appropriate oracles. It is the purpose of this note to investigate the role of phase information in the interaction between a quantum computer and an oracle. The answers provided by a quantum oracle contain both amplitude and phase information. In [2] it is assumed that the oracle increments the phase of the wave function by the same amount for all queries. However, there is no physical reason for this choice. If we allow for full use of phase information a quantum oracle can transmit information to a quantum computer which is inaccessible to a classical computer equipped with the same oracle. The conventional computer science definition of an oracle is a set X, X ⊆ Σ∗ where Σ∗ is the set of all finite bit strings. A classical realization of an oracle is a device which when fed a string x ∈ Σ∗ (the query) returns a ‘1’ if x ∈ X and ‘0’ otherwise. ∗This research was partially funded by the National Science Foundation Grant DMR-9311580.