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
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?
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
Classical Algorithm: Analysis The state for which f()=1 K needs to be np before this prob exceeds p
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
Unitariness Constraints Diagonals A, non- diagonals B
Unitariness Constraints Diagonal Matrix, so clearly Unitary