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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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Presentation Transcript
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:

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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