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| i | i | i | i = x ,
√2 ··· √2 √2b | i been shownthat quantum walk based algorithms
Xx
can offer quadratical speedup for certain Markov
x=x x ,x =0,1,
1 ··· b j chain based algorithms (Magniez et al. (2011);
where the sum is over all possible 2b values of x, Richter (2007); Szegedy (2004), and Wocjan and
and then measure the obtained qubits and yield Abeyesinghe (2008)).
21
QUANTUMCOMPUTATIONAND QUANTUMINFORMATION
Finally we point out that quantum computers are ACKNOWLEDGMENTS
wonderful but it is difficult to build quantum com-
Wang’s research was supported in part by NSF
puters with present technology. To build a quantum
GrantDMS-10-05635.HethankseditorDavidMadi-
computer the physical apparatus must satisfy re-
gan and two anonymous referees for helpful com-
quirementsthatthequantumsystemrealized qubits
ments and suggestions which led to significant im-
needs to be well isolated in order to retain its quan-
provements in both substance and the presentation
tum properties and at the same time the quantum
of the paper.
systemhastobeaccessible sothatthequbitscanbe
manipulated to perform computations and measure
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