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This text delves into Quantum Searching and its related algorithms, including Fixed Point Searching. Understand how the Quantum Search Algorithm outperforms classical methods and achieves exhaustively precise results. Dive into the intricacies of amplitude amplification and its impact on search efficiency.
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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)
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
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
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」).
Quantum Search Algorithm • Encode N states with log2N qubits. • Start with all qubits in 0 state. • Apply the following operations: Observe the state.
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.
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.
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.
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
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
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.”