1 / 14

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)

onawa
Download Presentation

Grover’s Algorithm: Single Solution

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related