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Quantum Speed-Up in Database Search

This article explores the simplest example of quantum speed-up in database search, specifically focusing on Grover's algorithm. The classical unstructured database search takes N-1 steps in the worst case, while the quantum search algorithm can find the desired item in just 1 step on average. The article also discusses the application of quantum mechanics in database search and the mapping of probability theory to Hamiltonian mechanics.

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Quantum Speed-Up in Database Search

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  1. Database search: the simplest example of quantum speed-up Armen Allahverdyan (Yerevan Physics Institute) -- Classical unstructured database search -- Reminder on quantum mechanics -- Quantum (Grover's) search

  2. Classical unstructured database search N items, N-1 identical, 1 special item to be found selector function Worst case: search takes N-1 steps. Best case: 1 step. In average: (N-1)/2 steps probability to find the red ball randomly is 1/N

  3. Databases are normally structured, but the unstructured assumption is still reasonable Time prior information (structure)

  4. Lot of prior information (structure); the red ball is heavier. The balls are arranged into regular structure and can be easily divided into equal parts search time

  5. Quantum mechanics = waves+ Hamiltonian mechanics + probability theory probability

  6. Mapping to Hamiltonian of N-level quantum system Hamiltonian is unknown E is a known parameter

  7. Interaction Hamiltonian and initial state known interaction Hamiltonian switched on at time zero is a known vector Schroedinger equation We want:

  8. Estimating the characteristic time via short-time expansion amplitude determines the change pure phase random known, controlled base

  9. averages characteristic time

  10. Exact solution of the Schroedinger equation motion sticks to two-dimensional Hilbert space two linear equations

  11. characteristic time quantum motion is determined by amplitudes, not by probabilities supersposition principle

  12. Alternative interaction Hamiltonians does not depend on N but you cannot apply this in search problems The simplest Hamiltonian is optimal at least for one-body physics

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