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# Grover’s Algorithm: Single Solution PowerPoint PPT Presentation

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)

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)

• 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:

Unitary operator describing 2-4:

### 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

• Black box limitations

• Physical implementation (classical memory) implementations