Quantum Counting

  title={Quantum Counting},
  author={Gilles Brassard and Peter H{\o}yer and Alain Tapp},
We study some extensions of Grover’s quantum searching algorithm. First, we generalize the Grover iteration in the light of a concept called amplitude amplification. Then, we show that the quadratic speedup obtained by the quantum searching algorithm over classical brute force can still be obtained for a large family of search problems for which good classical heuristics exist. Finally, as our main result, we combine ideas from Grover’s and Shor’s quantum algorithms to perform approximate… Expand
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Implementation of quantum search algorithm using classical Fourier optics.
An experiment on Grover's quantum search algorithm shows that classical waves can search a N-item database as efficiently as quantum mechanics can, although the lack of quantum entanglement limits the size of the database. Expand
From Monte Carlo to quantum computation
  • S. Heinrich
  • Computer Science, Physics
  • Math. Comput. Simul.
  • 2003
A short introduction to the basic ideas of quantum computing is given and connections to the Monte Carlo methology are discussed, to compare the optimal error rates of quantum algorithms to those of classical deterministic and randomized algorithms. Expand
On the Complexity of Searching for a Maximum of a Function on a Quantum Computer
older class on a quantum computer. We show matching lower and upper bounds on the complexity of this problem. We prove upper bounds by constructing an algorithm that uses a pre-existing quantumExpand
Quantum Counting: Algorithm and Error Distribution
Counting is one of the most basic procedures in mathematics and statistics. In statistics literature it is usually done via the proportion estimation method. In this article we manifest a radicallyExpand
On the Complexity of Searching for a Maximum of a Function on a Quantum Computer
It is shown that quantum computation yields a quadratic speed-up over deterministic and randomized algorithms. Expand
The Generalized Quantum Database Search Algorithm
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A lower bound on the efficiency of any possible quantum database searching algorithm is provided and it is shown that Grover''s algorithm nearly comes within a factor 2 of being optimal in terms of the number of probes required in the table. Expand
A Quantum Jump in Computer Science
This paper surveys some of the most striking new applications of quantum mechanics to computer science and some are still theoretical but others have been implemented. Expand
An exact quantum polynomial-time algorithm for Simon's problem
  • G. Brassard, P. Høyer
  • Mathematics, Physics
  • Proceedings of the Fifth Israeli Symposium on Theory of Computing and Systems
  • 1997
It is shown that there is a decision problem that can be solved in exact quantum polynomial time, which would require expected exponential time on any classical bounded-error probabilistic computer if the data is supplied as a black box. Expand
Quantum Physics and Computers
Abstract Recent theoretical results confirm that quantum theory provides the possibility of new ways of performing efficient calculations. The most striking example is the factoring problem. It hasExpand
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Quantum computers use the quantum interference of different computational paths to enhance correct outcomes and suppress erroneous outcomes of computations. A common pattern underpinning quantumExpand
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
  • P. Shor
  • Computer Science, Mathematics
  • SIAM Rev.
  • 1999
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Quantum Mechanics Helps in Searching for a Needle in a Haystack
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It is proved that relative to an oracle chosen uniformly at random with probability 1 the class $\NP$ cannot be solved on a quantum Turing machine (QTM) in time $o(2^{n/2})$. Expand
Quantum Database Search by a Single Query
A quantum mechanical algorithm that can search a database by a single query when the number of solutions is more than a quarter using the generalized Grover operator of arbitrary phase is given. Expand
Quantum Database Searching by a Single Query
In this paper we give a quantum mechanical algorithm that can search a database by a single query, when the number of solutions is more than a quarter. It utilizes modified Grover operator ofExpand