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lecture_11 - Lecture 11 Quantum Random Walks Lecturer Peter...

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Lecture 11 : Quantum Random Walks Lecturer: Peter Shor Scribe: Isaac Kim 1 Quantum Random Walks Exponential speedups on contrived problems Childs et al. speedups on some applicable problems Ambainis’s algorithm for element distinctness 2 Grover’s Algorithm We have N elements One of the are ‘marked’ Find it! Classically : O ( N ) Quantum Mechanically : O ( N ) Strategy Use two operations i = i where i is the marked one, G j = i = j G | −| N | | j ∀ ± 1 N : ψ = M | ψ ( M = 2 ψ I ) j → | | ψ ² | | j =1 Start in | ψ π 4 Perform ( MG ) t for t N = Why does it work? The state stays in a subspace generated by | ψ , | i .
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2 Isaac Kim 3 Generalization Suppose you have a N × N grid. We will use following operations 1. Move to adjacent vertex 2. Ask “Is this vertex marked?” For N × N grid, there is O ( N log N ) quantum algorithm. For dim 3 grids, O (
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This note was uploaded on 02/07/2012 for the course MAS 6.443J taught by Professor Petershor during the Spring '06 term at MIT.

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lecture_11 - Lecture 11 Quantum Random Walks Lecturer Peter...

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