Grover

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Grovers AlgorithmSingle Solution

• By Michael Kontz

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Application

• Grovers algorithm can identify an item from a
  list of N elements in
• Whats this good for?
• Unstructured database search
• (virtual database)
• breaking DES (Data Encryption Standard)
• SAT (Satisfyability of boolean formula)
• map coloring with 4 colors

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Application DES

• clear text key ciphertext
• attackatdawn 3726495784 ojbevjewbvv
• 56-bit key
• Best classical algorithm
• 36 quadrillion
• Grovers algorithm
• 118 million

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Amplitude Amplification

• Overview
• Start in an initial state that is equally every
  state
• Over time (iterations) amplify amplitude of
  solution
• Measure (collapse system) when amplitutde2 is
  greater than 0.5 (50)

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Initial State

• Hadamard Gate
• Steps N 2n
• begin in state
• transform into equal superposition of all states
  using Hadamard

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Oracle

• Oracle picks out which state to amplify
• Black box
• Oracle is unitary operator UO

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Oracle

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

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Algorithm

• Setup initial state
• Repeat these 4 steps times
• Measure answer

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Inversion about mean

• Unitary operator describing phase shift

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Inversion about mean

• Unitary operator describing 2-4

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Inversion about mean
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Complexity O(sqrt(N))

• How many calls to oracle does it take to achieve
  amplitude2 gt 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
  amplitude2 gt 0.5

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Limitations

• Black box limitations
• Physical implementation (classical memory)
  implementations