dorsal/arxiv
View SchemaQuantum Search of Spatial Regions
| Authors | Scott Aaronson, Andris Ambainis |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0303041 |
| URL | https://arxiv.org/abs/quant-ph/0303041 |
Abstract
Can Grover's algorithm speed up search of a physical region - for example a 2-D grid of size sqrt(n) by sqrt(n)? The problem is that sqrt(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(sqrt n) for d at least 3, or O((sqrt n)(log n)^(3/2)) 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(sqrt(n))-qubit communication protocol for the disjointness problem, which improves an upper bound of Hoyer and de Wolf and matches a lower bound of Razborov.
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"date_created": "2026-03-02T18:01:56.076000Z",
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"abstract": "Can Grover\u0027s algorithm speed up search of a physical region - for example a\n2-D grid of size sqrt(n) by sqrt(n)? The problem is that sqrt(n) time seems to\nbe needed for each query, just to move amplitude across the grid. Here we show\nthat this problem can be surmounted, refuting a claim to the contrary by\nBenioff. In particular, we show how to search a d-dimensional hypercube in time\nO(sqrt n) for d at least 3, or O((sqrt n)(log n)^(3/2)) for d=2. More\ngenerally, we introduce a model of quantum query complexity on graphs,\nmotivated by fundamental physical limits on information storage, particularly\nthe holographic principle from black hole thermodynamics. Our results in this\nmodel include almost-tight upper and lower bounds for many search tasks; a\ngeneralized algorithm that works for any graph with good expansion properties,\nnot just hypercubes; and relationships among several notions of `locality\u0027 for\nunitary matrices acting on graphs. As an application of our results, we give an\nO(sqrt(n))-qubit communication protocol for the disjointness problem, which\nimproves an upper bound of Hoyer and de Wolf and matches a lower bound of\nRazborov.",
"arxiv_id": "quant-ph/0303041",
"authors": [
"Scott Aaronson",
"Andris Ambainis"
],
"categories": [
"quant-ph",
"gr-qc"
],
"title": "Quantum Search of Spatial Regions",
"url": "https://arxiv.org/abs/quant-ph/0303041"
},
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