Searching using Quantum Rules
System State is captured by a  Probability                        Our Two Worlds  Distribution                            ...
What does this mean for    computation?
Simple Search• Given n items, A[1..n]• Exactly one satisfies f()=1  – all others have f()=0• Find the unique object for wh...
Classical Algorithmx  Random item in 1..nRepeat   –   – If f(x)=1      • Stay at x   – Else      • Choose another (never ...
Classical Algorithm: Analysis                The state for                 which f()=1                           K needs t...
Grover’s Quantum Algorithmx  Random item in 1..nRepeat   – If f(x)=1         • then stay at x with amplitude -1         •...
Unitariness Constraints                     Diagonals A, non-                        diagonals B
Unitariness Constraints        Diagonal Matrix, so          clearly Unitary
Algorithm Analysis                 k
State Progression                       For n=4, one                    iteration suffices!                     With certa...
State Progression in General            The state for             which f()=1                            What progress hav...
Progress per Iteration   Dot Product
Geometry           Destination           State Vector            [0 0……1]
Wrapping Up
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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

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