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A new quantum lower bound method,: with applications to direct product theorems and time-space tradeoffs

Published: 21 May 2006 Publication History

Abstract

We give a new version of the adversary method for proving lower bounds on quantum query algorithms. The new method is based on analyzing the eigenspace structure of the problem at hand. We use it to prove a new and optimal strong direct product theorem for 2-sided error quantum algorithms computing k independent instances of a symmetric Boolean function: if the algorithm uses significantly less than k times the number of queries needed for one instance of the function, then its success probability is exponentially small in k. We also use the polynomial method to prove a direct product theorem for 1-sided error algorithms for k threshold functions with a stronger bound on the success probability. Finally, we present a quantum algorithm for evaluating solutions to systems of linear inequalities, and use our direct product theorems to show that the time-space tradeoff of this algorithm is close to optimal.

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  • (2024)An Efficient Quantum Parallel Repetition Theorem and ApplicationsProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649603(1478-1487)Online publication date: 10-Jun-2024
  • (2008)Direct product theorems for classical communication complexity via subdistribution boundsProceedings of the fortieth annual ACM symposium on Theory of computing10.1145/1374376.1374462(599-608)Online publication date: 17-May-2008
  • (2008)The Multiplicative Quantum AdversaryProceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity10.1109/CCC.2008.9(237-248)Online publication date: 22-Jun-2008
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  1. A new quantum lower bound method,: with applications to direct product theorems and time-space tradeoffs

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      Bruce E. Litow

      Ambainis et al. present a new adversary method for deriving lower bounds on the number of queries needed in quantum computation-based direct product evaluation of symmetric Boolean functions. Previously, lower bounds for quantum computation have been obtained for specific Boolean functions, rather than for a family of functions. The direct product problem is the evaluation in parallel of the values of a function on k independent inputs. The lower bound is obtained by demonstrating a success probability that is exponentially small in k. The authors derive time/space tradeoffs for evaluating linear inequalities as an application of their symmetric function result. The paper also has a section on a different lower bound technique using polynomial degree bounding results from the literature. An interesting feature of the exposition is a readable appendix in which the eigenspace structures induced by various unitary operators are analyzed in a series of lemmas. These may be of use even to researchers doing computational linear algebra outside the quantum computation framework. The two main results are for 1-sided and 2-sided success probabilities. The 1-sided case amounts to insisting that all 1-bits (this is a convention) are correct. Intuition suggests, and the results bear out, that the success probability for the 1-sided case is smaller than that for the 2-sided case. Using their linear subspace structure technique for adversary arguments, the authors derive a 2-sided success probability upper bound for symmetric function f of 2Ω(k), where at most α.k.Q2(f) 2-sided (so Q2) quantum queries are permitted. The value Q2(f) depends heavily on the function f, and α is a positive constant. For a t-threshold function f (where f (x) = 0 if the Hamming weight of x is less than t, and f(x) = 1 otherwise), the authors derive a 1-sided upper bound of 2Ω(k.t). Note the presence of the threshold t in the exponent, which reflects the fact that each 1-bit must be correct. The 1-sided result uses polynomial degree bounding. Online Computing Reviews Service

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      cover image ACM Conferences
      STOC '06: Proceedings of the thirty-eighth annual ACM symposium on Theory of Computing
      May 2006
      786 pages
      ISBN:1595931341
      DOI:10.1145/1132516
      Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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      Publication History

      Published: 21 May 2006

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      Author Tags

      1. direct product theorems
      2. lower bounds
      3. quantum computing
      4. time-space tradeoffs

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      STOC06: Symposium on Theory of Computing
      May 21 - 23, 2006
      WA, Seattle, USA

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      Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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      View all
      • (2024)An Efficient Quantum Parallel Repetition Theorem and ApplicationsProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649603(1478-1487)Online publication date: 10-Jun-2024
      • (2008)Direct product theorems for classical communication complexity via subdistribution boundsProceedings of the fortieth annual ACM symposium on Theory of computing10.1145/1374376.1374462(599-608)Online publication date: 17-May-2008
      • (2008)The Multiplicative Quantum AdversaryProceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity10.1109/CCC.2008.9(237-248)Online publication date: 22-Jun-2008
      • (2008)A Direct Product Theorem for DiscrepancyProceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity10.1109/CCC.2008.25(71-80)Online publication date: 22-Jun-2008

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