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30/06/2017, 15:00 — 16:00 — Room P9, Mathematics Building

Daniel Reitzner, *Slovak Academy of Sciences*

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Navigating a Maze Using Quantum-Walk Searches

Quantum searches are used to localize a specific element within the Hilbert space. Usually such element is a base state, or a small subspace of the Hilbert space. We expand this set by searching for a path in a maze between two specific vertices using quantum walks. A particular type of maze we use is a chain of connected stars. We show, that having $M$ stars, each of them containing $N$ spikes, we can find the whole path in $O(M N^{1/2})$ steps. Standard choice of the initial state in a large, albeit phase-modulated superposition leads to the usual Grover search. However, such state is usually hard to prepare and a choice of a localized state is preferred. In such case we show, that it is possible to find the path with the same efficiency (up to a multiplicative constant) successively, by starting on the first star and uncovering successive stars by short searches of $O(N^{1/2})$ steps.