Article

Quantum lower bound for the collision problem

Author:

Scott AaronsonAuthors Info & Claims

STOC '02: Proceedings of the thiry-fourth annual ACM symposium on Theory of computing

Pages 635 - 642

https://doi.org/10.1145/509907.509999

Published: 19 May 2002 Publication History

Abstract

(MATH) The collision problem is to decide whether a function X: { 1,…,n} → { 1, …,n} is one-to-one or two-to-one, given that one of these is the case. We show a lower bound of Ω(n^1/5) on the number of queries needed by a quantum computer to solve this problem with bounded error probability. The best known upper bound is O(n^1/3), but obtaining any lower bound better than Ω(1) was an open problem since 1997. Our proof uses the polynomial method augmented by some new ideas. We also give a lower bound of Ω(n^1/7) for the problem of deciding whether two sets are equal or disjoint on a constant fraction of elements. Finally we give implications of these results for quantum complexity theory.

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Cited By

Yang GZhang JMohar BShinkar IO'Donnell R(2024)Communication Lower Bounds for Collision Problems via Density Increment ArgumentsProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649607(630-639)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649607
Ambainis ABelovs A(2023)An Exponential Separation between Quantum Query Complexity and the Polynomial DegreeProceedings of the conference on Proceedings of the 38th Computational Complexity Conference10.4230/LIPIcs.CCC.2023.24(1-13)Online publication date: 17-Jul-2023
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2023.24
Santos MMateus PPinto A(2022)Quantum Oblivious Transfer: A Short ReviewEntropy10.3390/e2407094524:7(945)Online publication date: 7-Jul-2022
https://doi.org/10.3390/e24070945
Show More Cited By

Index Terms

Quantum lower bound for the collision problem
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Ordinary differential equations
    2. Numerical analysis
      1. Computations on polynomials
      2. Numerical differentiation
2. Theory of computation
  1. Design and analysis of algorithms

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cover image ACM Conferences

STOC '02: Proceedings of the thiry-fourth annual ACM symposium on Theory of computing

May 2002

840 pages

ISBN:1581134959

DOI:10.1145/509907

Conference Chair:
John Reif
Duke University

Copyright © 2002 ACM.

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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SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

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New York, NY, United States

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Published: 19 May 2002

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STOC02: Symposium on the Theory of Computing

May 19 - 21, 2002

Quebec, Montreal, Canada

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STOC '02 Paper Acceptance Rate 91 of 287 submissions, 32%;

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

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Cited By

Yang GZhang JMohar BShinkar IO'Donnell R(2024)Communication Lower Bounds for Collision Problems via Density Increment ArgumentsProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649607(630-639)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649607
Ambainis ABelovs A(2023)An Exponential Separation between Quantum Query Complexity and the Polynomial DegreeProceedings of the conference on Proceedings of the 38th Computational Complexity Conference10.4230/LIPIcs.CCC.2023.24(1-13)Online publication date: 17-Jul-2023
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2023.24
Santos MMateus PPinto A(2022)Quantum Oblivious Transfer: A Short ReviewEntropy10.3390/e2407094524:7(945)Online publication date: 7-Jul-2022
https://doi.org/10.3390/e24070945
Aaronson S(2021)Open Problems Related to Quantum Query ComplexityACM Transactions on Quantum Computing10.1145/34885592:4(1-9)Online publication date: 21-Dec-2021
https://dl.acm.org/doi/10.1145/3488559
Aaronson SKothari RKretschmer WThaler JAceto L(2020)Quantum lower bounds for approximate counting via laurent polynomialsProceedings of the 35th Computational Complexity Conference10.4230/LIPIcs.CCC.2020.7(1-47)Online publication date: 28-Jul-2020
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2020.7
Grassi LNaya-Plasencia MSchrottenloher A(2018)Quantum Algorithms for the $$k$$-xor ProblemAdvances in Cryptology – ASIACRYPT 201810.1007/978-3-030-03326-2_18(527-559)Online publication date: 27-Oct-2018
https://doi.org/10.1007/978-3-030-03326-2_18
Bouland AChen LHolden DThaler JVasudevan P(2017)On the Power of Statistical Zero Knowledge2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2017.71(708-719)Online publication date: Oct-2017
https://doi.org/10.1109/FOCS.2017.71
Aaronson SBouland AFitzsimons JLee MSudan M(2016)The Space "Just Above" BQPProceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science10.1145/2840728.2840739(271-280)Online publication date: 14-Jan-2016
https://dl.acm.org/doi/10.1145/2840728.2840739
Wu WZhang HMao SWang H(2015)Quantum algorithm to find invariant linear structure of MD hash functionsQuantum Information Processing10.1007/s11128-014-0909-514:3(813-829)Online publication date: 1-Mar-2015
https://dl.acm.org/doi/10.1007/s11128-014-0909-5
Harlow DHayden P(2013)Quantum computation vs. firewallsJournal of High Energy Physics10.1007/JHEP06(2013)0852013:6Online publication date: 21-Jun-2013
https://doi.org/10.1007/JHEP06(2013)085
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