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

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)
• map coloring with 4 colors
Application: DES
• clear text + key = ciphertext
• “attackatdawn” + 3726495784 = “ojbevjewbvv”
• 56-bit key
• Best classical algorithm
• Grover’s algorithm
• 118 million
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
• Steps: N = 2^n
• begin in state
• transform into equal superposition of all states using Hadamard
Oracle
• Oracle picks out which state to amplify
• Black box:
• Oracle is unitary operator UO:
Oracle
• Conjugate oracle with Hadamard transforms so only changes phase (sign)
Algorithm
• Setup initial state
• Repeat these 4 steps times

Unitary operator describing phase shift: