12 research outputs found
Robust quantum spatial search
Quantum spatial search has been widely studied with most of the study
focusing on quantum walk algorithms. We show that quantum walk algorithms are
extremely sensitive to systematic errors. We present a recursive algorithm
which offers significant robustness to certain systematic errors. To search N
items, our recursive algorithm can tolerate errors of size O(1/\sqrt{\ln N})
which is exponentially better than quantum walk algorithms for which tolerable
error size is only O(\ln N/\sqrt{N}). Also, our algorithm does not need any
ancilla qubit. Thus our algorithm is much easier to implement experimentally
compared to quantum walk algorithms
Quantum walks on two-dimensional grids with multiple marked locations
The running time of a quantum walk search algorithm depends on both the
structure of the search space (graph) and the configuration of marked
locations. While the first dependence have been studied in a number of papers,
the second dependence remains mostly unstudied.
We study search by quantum walks on two-dimensional grid using the algorithm
of Ambainis, Kempe and Rivosh [AKR05]. The original paper analyses one and two
marked location cases only. We move beyond two marked locations and study the
behaviour of the algorithm for an arbitrary configuration of marked locations.
In this paper we prove two results showing the importance of how the marked
locations are arranged. First, we present two placements of marked
locations for which the number of steps of the algorithm differs by
factor. Second, we present two configurations of and
marked locations having the same number of steps and probability to
find a marked location
On Applying the Lackadaisical Quantum Walk Algorithm to Search for Multiple Solutions on Grids
Quantum computing holds the promise of improving the information processing
power to levels unreachable by classical computation. Quantum walks are heading
the development of quantum algorithms for searching information on graphs more
efficiently than their classical counterparts. A quantum-walk-based algorithm
that is standing out in the literature is the lackadaisical quantum walk. The
lackadaisical quantum walk is an algorithm developed to search two-dimensional
grids whose vertices have a self-loop of weight . In this paper, we address
several issues related to the application of the lackadaisical quantum walk to
successfully search for multiple solutions on grids. Firstly, we show that only
one of the two stopping conditions found in the literature is suitable for
simulations. We also demonstrate that the final success probability depends on
the space density of solutions and the relative distance between solutions.
Furthermore, this work generalizes the lackadaisical quantum walk to search for
multiple solutions on grids of arbitrary dimensions. In addition, we propose an
optimal adjustment of the self-loop weight for such scenarios of arbitrary
dimensions. It turns out the other fits of found in the literature are
particular cases. Finally, we observe a two-to-one relation between the steps
of the lackadaisical quantum walk and the ones of Grover's algorithm, which
requires modifications in the stopping condition. In conclusion, this work
deals with practical issues one should consider when applying the lackadaisical
quantum walk, besides expanding the technique to a wider range of search
problems.Comment: Extended version of the conference paper available at
https://doi.org/10.1007/978-3-030-61377-8_9 . 21 pages, 6 figure