Complex Quantum Queries

A quantum mechanical computer can search a database consisting of N items in elementary quantum queries (a query is defined as any question to the database to which a binary (YES/NO) answer is required; an elementary query is a query pertaining to only one of the N possible states). A classical computer would take at least O(N) elementary queries. This paper shows that a quantum mechanical algorithm that can query information relating to multiple states, can search the database in a single query. A classical algorithm will be limited to the information theoretic bound of at least queries (which it would achieve by using a binary search). 0. Background Quantum mechanical computers can carry out multiple operations on a single piece of hardware at the same time. An example of an algorithm that uses this parallelism is the one mentioned in the abstract above [Search] which searches an N item database for a single item in quantum queries, where each query pertains to only one of the N states. This was in some ways a surprising result, in some ways not so surprising. To those used to dealing with classical entities, this was surprising since there are N items to be searched, so how could the result be obtained faster than N steps? However, from a quantum mechanical point of view all N items are being simultaneously searched, so there is no obvious reason the results could not be obtained in a single query. By means of subtle reasoning about unitary transformations, [BBHT] show that quantum mechanical algorithms could not search faster than elementary queries because of fundamental limitations. In the algorithm of [Search], only elementary queries were permitted, i.e. if A, B, C & D be the possible states, one query could only ask a question about a single state, e.g. " Is A the marked state? " A related question is: how long would it take if the queries can be about multiple states, e.g. " Is either A or B the marked state? " From elementary information theoretic principles, it follows that the best a classical computer could do, by means of such queries , would be a binary search. This would take queries. This paper shows that in case the quantum computer can process information about multiple states, then it is possible to design an algorithm that can indeed search the entire database in a single …