## Quantum search of spatial regions (2005)

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- [theoryofcomputing.org]
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- [arxiv.org]
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### Other Repositories/Bibliography

Venue: | THEORY OF COMPUTING |

Citations: | 57 - 8 self |

### BibTeX

@INPROCEEDINGS{Aaronson05quantumsearch,

author = {Scott Aaronson and Andris Ambainis},

title = {Quantum search of spatial regions},

booktitle = {THEORY OF COMPUTING},

year = {2005},

pages = {200--209},

publisher = {}

}

### Years of Citing Articles

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### Abstract

Can Grover’s algorithm speed up search of a physical region—for example a 2-D grid of size √ n × √ n? The problem is that √ n time seems to be needed for each query, just to move amplitude across the grid. Here we show that this problem can be surmounted, refuting a claim to the contrary by Benioff. In particular, we show how to search a d-dimensional hypercube in time O ( √ n) for d ≥ 3, or O ( √ nlog 5/2 n) for d = 2. More generally, we introduce a model of quantum query complexity on graphs, motivated by fundamental physical limits on information storage, particularly the holographic principle from black hole thermodynamics. Our results in this model include almost-tight upper and lower bounds for many search tasks; a generalized algorithm that works for any graph with good expansion properties, not just hypercubes; and relationships among several notions of ‘locality’ for unitary matrices acting on graphs. As an application of our results, we give an O (√ n)-qubit communication protocol for the disjointness problem, which improves an upper bound of Høyer and de Wolf and matches a lower bound of Razborov.