dorsal/arxiv
View SchemaQuantum computation from a quantum logical perspective
| Authors | Jeffrey Bub |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0605243 |
| URL | https://arxiv.org/abs/quant-ph/0605243 |
Abstract
It is well-known that Shor's factorization algorithm, Simon's period-finding algorithm, and Deutsch's original XOR algorithm can all be formulated as solutions to a hidden subgroup problem. Here the salient features of the information-processing in the three algorithms are presented from a different perspective, in terms of the way in which the algorithms exploit the non-Boolean quantum logic represented by the projective geometry of Hilbert space. From this quantum logical perspective, the XOR algorithm appears directly as a special case of Simon's algorithm, and all three algorithms can be seen as exploiting the non-Boolean logic represented by the subspace structure of Hilbert space in a similar way. Essentially, a global property of a function (such as a period, or a disjunctive property) is encoded as a subspace in Hilbert space representing a quantum proposition, which can then be efficiently distinguished from alternative propositions, corresponding to alternative global properties, by a measurement (or sequence of measurements) that identifies the target proposition as the proposition represented by the subspace containing the final state produced by the algorithm.
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"abstract": "It is well-known that Shor\u0027s factorization algorithm, Simon\u0027s period-finding\nalgorithm, and Deutsch\u0027s original XOR algorithm can all be formulated as\nsolutions to a hidden subgroup problem. Here the salient features of the\ninformation-processing in the three algorithms are presented from a different\nperspective, in terms of the way in which the algorithms exploit the\nnon-Boolean quantum logic represented by the projective geometry of Hilbert\nspace. From this quantum logical perspective, the XOR algorithm appears\ndirectly as a special case of Simon\u0027s algorithm, and all three algorithms can\nbe seen as exploiting the non-Boolean logic represented by the subspace\nstructure of Hilbert space in a similar way. Essentially, a global property of\na function (such as a period, or a disjunctive property) is encoded as a\nsubspace in Hilbert space representing a quantum proposition, which can then be\nefficiently distinguished from alternative propositions, corresponding to\nalternative global properties, by a measurement (or sequence of measurements)\nthat identifies the target proposition as the proposition represented by the\nsubspace containing the final state produced by the algorithm.",
"arxiv_id": "quant-ph/0605243",
"authors": [
"Jeffrey Bub"
],
"categories": [
"quant-ph"
],
"title": "Quantum computation from a quantum logical perspective",
"url": "https://arxiv.org/abs/quant-ph/0605243"
},
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