Quantum search algorithms on the hypercube

Details Quantum search algorithms on the hypercube Quantum search algorithms on the hypercube

2009-06-17

arxiv.org/abs/0906.3094v1

J. Phys. A: Math. Theor. 42 (2009) 085303

Physics

Quantum Physics

Scientific paper

10.1088/1751-8113/42/8/085303

We investigate a set of discrete-time quantum search algorithms on the n-dimensional hypercube following a proposal by Shenvi, Kempe and Whaley. We show that there exists a whole class of quantum search algorithms in the symmetry reduced space which perform a search of a marked vertex in time of order $\sqrt{N}$ where $N = 2^n$, the number of vertices. In analogy to Grover's algorithm, the spatial search is effectively facilitated through a rotation in a two-level sub-space of the full Hilbert space. In the hypercube, these two-level systems are introduced through avoided crossings. We give estimates on the quantum states forming the 2-level sub-spaces at the avoided crossings and derive improved estimates on the search times.

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