Quantum Direct Product Theorems for Symmetric Functions and Time-Space Tradeoffs

A direct product theorem upper-bounds the overall success probability of algorithms for computing many independent instances of a computational problem. We prove a direct product theorem for 2-sided error algorithms for symmetric functions in the setting of quantum query complexity, and a stronger direct product theorem for 1-sided error algorithms for threshold functions. We also present a quantum algorithm for deciding systems of linear inequalities, and use our direct product theorems to show that the time-space tradeoff of this algorithm is close to optimal. Institute for Quantum Computing and Department of Combinatorics and Optimization, University of Waterloo. Supported by NSERC, ARO, CIAR and IQC University Professorship. Supported in part by the EU fifth framework project RESQ, IST-2001-37559. Supported by a Veni grant from the Netherlands Organization for Scientific Research (NWO) and by the EU fifth framework project RESQ, IST-2001-37559.