Searching – quantum & classical Quantum Searching Fixed Point Searching The search algorithm combines the two main building blocks for quantum algorithms---fast transforms and amplitude amplification---and is deceptively simple. - David Meyer (Three views of the search algorithm) Quantum Searching & Related Algorithms Lov K. Grover, Bell Labs, Alcatel-Lucent

Classical Searching out of 5 items NO ITEM 2 ITEM 3ITEM 4ITEM 5 NO ITEM 3ITEM 4ITEM 5 ITEM 1 NO ITEM 2 ITEM 4ITEM 5ITEM 1 AHA! ITEM 2 ITEM 3ITEM 5ITEM 1

Quantum Mechanical Search NO AHA! NO AHA! NO Design a scheme so that chance of being in state is high. AHA! Now if the system is observed, there is a high probability of observing state. AHA!

Search – Quantum & Classical In amplitude amplification, amplitude in target state is amplified. (after  iterations, the probability of success is |sin(2  U ts ) 2 |). In classical searching probabilities in non-target states is reduced (e.g. after  iterations, the probability of success is 1- (1-|Uts| 2 )  」 .

Quantum Search Algorithm Encode N states with log 2 N qubits. Start with all qubits in 0 state. Apply the following operations: Observe the state.

Optimality of quantum search algorithm Given the following block - f(x) 0/1 We are allowed to hook up O(log N) hardware. Problem - find the single point at which f(x) ≠ 0. Quantum search algorithm is best possible algorithm for exhaustive searching. - Chris Zalka, Phys. Rev. A, 1999 Classically we need N steps. Quantum mechanically, we need only √N steps. However, only optimal for exhaustive search of 1 in N items.

Quantum searching amidst uncertainty Quantum search algorithm is optimal only if number of solutions is known. Puzzle - Find a solution if the number of solutions is either 1 or 2 with equal probability. (Only one observation allowed) Fixed point searching converges to 1. ½+½(1-(½)  t/4 ) Maximum success probability = 3/4 ½(sin 2 (t)+sin 2 (2t))

Fixed Point Quantum Searching Fixed point Target state of (standard) quantum search Fixed point – point of monotonic convergence (no overshoot). Fixed points achieved by 1. Using measurements 2. Iterating with slightly different unitary operations in different iterations. Iterative quantum procedures cannot have fixed points (Reason – Unitary transformations have eigenvalues of modulus unity so inherently periodic).

If |V ts | 2 = 1-  denote  /3 phase shift of t & s state by R t & R s. |VR s V † R t V| ts 2 = 1-   | V(R s V † R t V)(R s V † R † t V )(R † s V † R t V)(R s V † R t V)| ts 2 = 1-   Non-periodic sequence and can hence have fixed-points Slightly different operations in different iterations

Error correction - idea U takes us to within  of the target state. | | 2 =1-  then UR s U † R t U takes us to within  3 of target | | 2 =1-   Can cancel errors in any unitary U by UR s U † R t U: - need to run U twice and U † once, with same errors. - need to be able to do R s & R t  |s> |t> U|s>  UR s U † R t U|s> |t> |s>

Quantum search Database search & function inversion Scheduling Problems Collision problem & Element Distinctness Precision Measurements Pendulum Modes Moving Particles in a Harmonic oscillator Confocal Resonator Design. “A good idea finds application in contexts beyond where it was originally conceived.”