Proceedings 41st Annual Symposium on Foundations of Computer Science (2000)
Redondo Beach, California
Nov. 12, 2000 to Nov. 14, 2000
J. Radhakrishnan , Sch. of Technol. & Comput. Sci., Tata Inst. of Fundamental Res., Mumbai, India
P. Sen , Sch. of Technol. & Comput. Sci., Tata Inst. of Fundamental Res., Mumbai, India
S. Venkatesh , Sch. of Technol. & Comput. Sci., Tata Inst. of Fundamental Res., Mumbai, India
Studies the quantum complexity of the static set membership problem: given a subset S (|S|/spl les/n) of a universe of size m(/spl Gt/n), store it as a table, T:(0,1)/sup r//spl rarr/(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 a few bit probes. This problem was considered by H. Buhrman et al. (2000), who showed lower and upper bounds for this problem in the classical deterministic and randomised 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/sub T/ that takes the basis state (y,b) to the basis state |y,b/spl oplus/T(y)<. The query algorithm is allowed to apply O/sub T/ on any superposition of basis states. We show tradeoff results between the space (defined as 2/sup r/) and the number of probes (oracle calls) in this model. Our results show that the lower bounds shown by Buhrman et al. 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.
quantum computing; computational complexity; set theory; probes; linear algebra; query processing; quantum complexity; static set membership problem; bit table; query answering; lower bounds; upper bounds; quantum bit-probe model; black-box unitary transform; oracle calls; query algorithm; basis state superposition; space-probe tradeoff; linear algebra
J. Radhakrishnan, S. Venkatesh and P. Sen, "The quantum complexity of set membership," Proceedings 41st Annual Symposium on Foundations of Computer Science(FOCS), Redondo Beach, California, 2000, pp. 554.