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Grover’s Algorithm: Single Solution

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Grover’s Algorithm:Single Solution

By Michael Kontz

- 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

- clear text + key = ciphertext
- “attackatdawn” + 3726495784 = “ojbevjewbvv”
- 56-bit key
- Best classical algorithm
- 36 quadrillion

- Grover’s algorithm
- 118 million

- 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%)

- Hadamard Gate
- Steps: N = 2^n
- begin in state
- transform into equal superposition of all states using Hadamard

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

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

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

Unitary operator describing phase shift:

Unitary operator describing 2-4:

Inversion about mean

- 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

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