Quantum Searching & Related Algorithms Lov K. Grover, Bell Labs, Alcatel-Lucent

1. Quantum Searching & Related AlgorithmsLov K. Grover, Bell Labs, Alcatel-Lucent • 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)

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

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

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

5. Quantum Search Algorithm • Encode N states with log2N qubits. • Start with all qubits in 0 state. • Apply the following operations: Observe the state.

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

7. 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) ½+½(1-(½)pt/4) ½(sin2(t)+sin2(2t)) Maximum success probability = 3/4 Fixed point searching converges to 1.

8. Fixed point • Target state of (standard) quantum search Fixed Point Quantum Searching • Fixed point – point of monotonic convergence (no overshoot). • Iterative quantum procedures cannot have fixed points(Reason – Unitary transformations have eigenvalues of modulus unity so inherently periodic). • Fixed points achieved by 1. Using measurements2. Iterating with slightly different unitary operations in different iterations.

9. Slightly different operations in different iterations • If|Vts|2 = 1-d, denote p/3 phase shift of t & s state by Rt & Rs. • |VRsV †RtV|ts2 = 1-d3 | V(RsV†RtV)(RsV†R†tV )(R†sV†RtV)(RsV†RtV)|ts2= 1-d9 • Non-periodic sequence and can hence have fixed-points

10. e |t> U|s> |s> e3 |t> URsU†RtU|s> |s> Error correction - idea • U takes us to within e of the target state. |<t|U|s>|2 =1- e then URsU†RtU takes us to within e3of target |<t|URsU†RtU|s>|2=1-e3 • Can cancel errors in any unitary U by URsU†RtU: - need to run U twice and U† once, with same errors.- need to be able to do Rs & Rt

11. 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.”