The Quantum Complexity of Set Membership

Abstract. We study the quantum complexity of the static set membership problem: given a subset S (|S| ≤ n ) of a universe of size m ( >> n ), store it as a table, T: {0,1}^r --> {0,1} , of bits so that queries of the form ``Is x in S ?'' can be answered. The goal is to use a small table and yet answer queries using few bit probes. This problem was considered recently by Buhrman et al. [BMRV], who showed lower and upper bounds for this problem in the classical deterministic and randomized models. In this paper we formulate this problem in the ``quantum bit probe model''. We assume that access to the table T is provided by means of a black box (oracle) unitary transform O _T that takes the basis state | y,b > to the basis state | y,b

\oplus

T(y) > . The query algorithm is allowed to apply O _T on any superposition of basis states.

We show tradeoff results between space (defined as 2 ^r ) and number of probes (oracle calls) in this model. Our results show that the lower bounds shown in [BMRV] for the classical model also hold (with minor differences) in the quantum bit probe model. These bounds almost match the classical upper bounds. Our lower bounds are proved using linear algebraic arguments.