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Grover’s Algorithm: Single Solution

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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  1. Grover’s Algorithm:Single Solution By Michael Kontz

  2. 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

  3. Application: DES • clear text + key = ciphertext • “attackatdawn” + 3726495784 = “ojbevjewbvv” • 56-bit key • Best classical algorithm • 36 quadrillion • Grover’s algorithm • 118 million

  4. 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%)

  5. Initial State • Hadamard Gate • Steps: N = 2^n • begin in state • transform into equal superposition of all states using Hadamard

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

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

  8. Algorithm • Setup initial state • Repeat these 4 steps times • Measure answer

  9. Inversion about mean Unitary operator describing phase shift:

  10. Inversion about mean Unitary operator describing 2-4:

  11. Inversion about mean

  12. 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

  13. Limitations • Black box limitations • Physical implementation (classical memory) implementations

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