Efficient Discrete-Time Simulations of Continuous-Time Quantum Query Algorithms

Efficient Discrete-Time Simulations of Continuous-Time Quantum Query Algorithms

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Unformatted text preview: Efficient discrete-time simulations of continuous-time quantum query algorithms R. Cleve, 1,2, * D. Gottesman, 2, † M. Mosca, 1,2, ‡ R. D. Somma, 2, § and D. L. Yonge-Mallo 1, ¶ 1 Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada 2 Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada (Dated: November 26, 2008) The continuous-time query model is a variant of the discrete query model in which queries can be interleaved with known operations (called “driving operations”) continuously in time. Interesting algorithms have been discovered in this model, such as an algorithm for evaluating nand trees more efficiently than any classical algorithm. Subsequent work has shown that there also exists an efficient algorithm for nand trees in the discrete query model; however, there is no efficient conversion known for continuous-time query algorithms for arbitrary problems. We show that any quantum algorithm in the continuous-time query model whose total query time is T can be simulated by a quantum algorithm in the discrete query model that makes O ( T log T/ log log T ) ⊂ ˜ O ( T ) queries. This is the first upper bound that is independent of the driving operations (i.e., it holds even if the norm of the driving Hamiltonian is very large). A corol- lary is that any lower bound of T queries for a problem in the discrete-time query model immediately carries over to a lower bound of Ω( T log log T/ log T ) ⊂ ˜ Ω( T ) in the continuous-time query model. I. INTRODUCTION AND SUMMARY OF RESULTS In the query (a.k.a. black-box or oracle) model of computation, one is given a black box that computes the individual entries of an N-tuple, x = ( x ,x 1 ,...,x N- 1 ), and the goal is to compute some function of these values, making as few queries to the black-box as possible. Many quantum algorithms can be naturally viewed as algorithms in this model, including Shor’s factoring algorithm [1], whose primary component computes the periodicity of a periodic sequence x ,x 1 ,...,x N- 1 (technically, the sequence must be also be distinct within each period). Other examples are [2, 3, 4]. In the quantum query model, a (full) quantum query is a unitary operation Q x such that Q x | j i| b i = | j i| b ⊕ x j i , (1) for all j ∈ { , 1 ,...,N- 1 } and b from the set of values that entries of the N-tuple ranges over, and ⊕ can be set to the bit-wise exclusive-or. Queries are interleaved with other quantum operations that “drive” the computation. The query cost of an algorithm is the number of queries that it makes. The efficiency of the other operations, besides queries, is also of interest. An algorithm is deemed efficient if it is efficient in both counts....
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This note was uploaded on 02/07/2011 for the course PHYS 101 taught by Professor Aster during the Spring '11 term at East Tennessee State University.

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Efficient Discrete-Time Simulations of Continuous-Time Quantum Query Algorithms

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