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# Searching using Quantum Rules

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Searching using Quantum Rules

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### Searching using Quantum Rules

1. 1. Searching using Quantum Rules
2. 2. System State is captured by a Probability Our Two Worlds Distribution Σ vi = 1 , 0<=vi<=1 T is stochastic (non-neg, col sums 1) Classical Measurement Quantum |v|2 = v†v = Σ |vi|2 = 1 System State is T is Unitary T†T = I captured by an Amplitude Distribution
3. 3. What does this mean for computation?
4. 4. Simple Search• Given n items, A[1..n]• Exactly one satisfies f()=1 – all others have f()=0• Find the unique object for which f()=1• How many f() calls are needed?
5. 5. Classical Algorithmx  Random item in 1..nRepeat – – If f(x)=1 • Stay at x – Else • Choose another (never previously picked) value for x, each possible value equi-probablyK times
6. 6. Classical Algorithm: Analysis The state for which f()=1 K needs to be np before this prob exceeds p
7. 7. Grover’s Quantum Algorithmx  Random item in 1..nRepeat – If f(x)=1 • then stay at x with amplitude -1 • else stay at x with amplitude 1 – Choose – this x with amplitude a – another value for x, each possible value with equal amplitude, say bK times
8. 8. Unitariness Constraints Diagonals A, non- diagonals B
9. 9. Unitariness Constraints Diagonal Matrix, so clearly Unitary
10. 10. Algorithm Analysis k
11. 11. State Progression For n=4, one iteration suffices! With certainity!!
12. 12. State Progression in General The state for which f()=1 What progress have we made?
13. 13. Progress per Iteration Dot Product
14. 14. Geometry Destination State Vector [0 0……1]
15. 15. Wrapping Up