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Quantum Approximate Counting with Nonadaptive Grover Iterations

Authors Ramgopal Venkateswaran, Ryan O'Donnell

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Ramgopal Venkateswaran
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Ryan O'Donnell
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA


The authors would like to thank Scott Aaronson and Patrick Rall for helpful communications.

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Ramgopal Venkateswaran and Ryan O'Donnell. Quantum Approximate Counting with Nonadaptive Grover Iterations. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 59:1-59:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


Approximate Counting refers to the problem where we are given query access to a function f : [N] → {0,1}, and we wish to estimate K = #{x : f(x) = 1} to within a factor of 1+ε (with high probability), while minimizing the number of queries. In the quantum setting, Approximate Counting can be done with O(min (√{N/ε}, √{N/K} / ε) queries. It has recently been shown that this can be achieved by a simple algorithm that only uses "Grover iterations"; however the algorithm performs these iterations adaptively. Motivated by concerns of computational simplicity, we consider algorithms that use Grover iterations with limited adaptivity. We show that algorithms using only nonadaptive Grover iterations can achieve O(√{N/ε}) query complexity, which is tight.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • quantum approximate counting
  • Grover search


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