dorsal/arxiv
View SchemaMore On Grover's Algorithm
| Authors | Ken Loo |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0502137 |
| URL | https://arxiv.org/abs/quant-ph/0502137 |
Abstract
The goals of this paper are to show the following. First, Grover's algorithm can be viewed as a digital approximation to the analog quantum algorithm proposed in "An Analog Analogue of a Digital Quantum Computation", by E. Farhi and S. Gutmann, Phys.Rev. A 57, 2403 - 2406 (1998), quant-ph/9612026. We will call the above analog algorithm the Grover-Farhi-Gutmann or GFG algorithm. Second, the propagator of the GFG algorithm can be written as a sum-over-paths formula and given a sum-over-path interpretation, i.e., a Feynman path sum/integral. We will use nonstandard analysis to do this. Third, in the semi-classical limit $\hbar\to 0$, both the Grover and the GFG algorithms (viewed in the setting of the approximation in this paper) must run instantaneously. Finally, we will end the paper with an open question. In "Semiclassical Shor's Algorithm", by P. Giorda, et al, Phys. Rev.A 70, 032303 (2004), quant-ph/0303037, the authors proposed building semi-classical quantum computers to run Shor's algorithm because the success probability of Shor's algorithm does not change much in the semi-classical limit. We ask the open questions: In the semi-classical limit, does Shor's algorithm have to run instantaneously?
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"abstract": "The goals of this paper are to show the following. First, Grover\u0027s algorithm\ncan be viewed as a digital approximation to the analog quantum algorithm\nproposed in \"An Analog Analogue of a Digital Quantum Computation\", by E. Farhi\nand S. Gutmann, Phys.Rev. A 57, 2403 - 2406 (1998), quant-ph/9612026. We will\ncall the above analog algorithm the Grover-Farhi-Gutmann or GFG algorithm.\nSecond, the propagator of the GFG algorithm can be written as a sum-over-paths\nformula and given a sum-over-path interpretation, i.e., a Feynman path\nsum/integral. We will use nonstandard analysis to do this. Third, in the\nsemi-classical limit $\\hbar\\to 0$, both the Grover and the GFG algorithms\n(viewed in the setting of the approximation in this paper) must run\ninstantaneously. Finally, we will end the paper with an open question. In\n\"Semiclassical Shor\u0027s Algorithm\", by P. Giorda, et al, Phys. Rev.A 70, 032303\n(2004), quant-ph/0303037, the authors proposed building semi-classical quantum\ncomputers to run Shor\u0027s algorithm because the success probability of Shor\u0027s\nalgorithm does not change much in the semi-classical limit. We ask the open\nquestions: In the semi-classical limit, does Shor\u0027s algorithm have to run\ninstantaneously?",
"arxiv_id": "quant-ph/0502137",
"authors": [
"Ken Loo"
],
"categories": [
"quant-ph"
],
"title": "More On Grover\u0027s Algorithm",
"url": "https://arxiv.org/abs/quant-ph/0502137"
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