Grover s algorithm single solution
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Grover’s Algorithm: Single Solution. By Michael Kontz. Application. Grover’s algorithm can identify an item from a list of N elements in What’s this good for? Unstructured database search (virtual database) breaking DES (Data Encryption Standard) SAT (Satisfyability of boolean formula)

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Grover’s Algorithm: Single Solution

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Grover s algorithm single solution

Grover’s Algorithm:Single Solution

By Michael Kontz


Application

Application

  • Grover’s algorithm can identify an item from a list of N elements in

  • What’s this good for?

    Unstructured database search

    (virtual database)

    • breaking DES (Data Encryption Standard)

    • SAT (Satisfyability of boolean formula)

    • map coloring with 4 colors


Application des

Application: DES

  • clear text + key = ciphertext

  • “attackatdawn” + 3726495784 = “ojbevjewbvv”

  • 56-bit key

  • Best classical algorithm

    • 36 quadrillion

  • Grover’s algorithm

    • 118 million


Amplitude amplification

Amplitude Amplification

  • Overview

    • Start in an initial state that is equally every state

    • Over time (iterations) amplify amplitude of solution

    • Measure (collapse system) when amplitutde^2 is greater than 0.5 (50%)


Initial state

Initial State

  • Hadamard Gate

  • Steps: N = 2^n

    • begin in state

    • transform into equal superposition of all states using Hadamard


Oracle

Oracle

  • Oracle picks out which state to amplify

  • Black box:

  • Oracle is unitary operator UO:


Oracle1

Oracle

  • Conjugate oracle with Hadamard transforms so only changes phase (sign)


Algorithm

Algorithm

  • Setup initial state

  • Repeat these 4 steps times

  • Measure answer


Inversion about mean

Inversion about mean

Unitary operator describing phase shift:


Inversion about mean1

Inversion about mean

Unitary operator describing 2-4:


Grover s algorithm single solution

Inversion about mean


Complexity o sqrt n

Complexity O(sqrt(N))

  • How many calls to oracle does it take to achieve amplitude^2 > 0.5?

    • assume all states but one have

    • other state is

    • each iteration

    • as long as

    • for large N this is true long enough for amplitude^2 > 0.5


Limitations

Limitations

  • Black box limitations

  • Physical implementation (classical memory) implementations


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