dorsal/arxiv
View SchemaThe Representation of Natural Numbers in Quantum Mechanics
| Authors | Paul Benioff |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0003063 |
| URL | https://arxiv.org/abs/quant-ph/0003063 |
| DOI | 10.1103/PhysRevA.63.032305 |
| Journal | Phys. Rev. A63, 032305 (2001) |
Abstract
This paper represents one approach to making explicit some of the assumptions and conditions implied in the widespread representation of numbers by composite quantum systems. Any nonempty set and associated operations is a set of natural numbers or a model of arithmetic if the set and operations satisfy the axioms of number theory or arithmetic. This work is limited to k-ary representations of length L and to the axioms for arithmetic modulo k^{L}. A model of the axioms is described based on states in and operators on an abstract L fold tensor product Hilbert space H^{arith}. Unitary maps of this space onto a physical parameter based product space H^{phy} are then described. Each of these maps makes states in H^{phy}, and the induced operators, a model of the axioms. Consequences of the existence of many of these maps are discussed along with the dependence of Grover's and Shor's Algorithms on these maps. The importance of the main physical requirement, that the basic arithmetic operations are efficiently implementable, is discussed. This conditions states that there exist physically realizable Hamiltonians that can implement the basic arithmetic operations and that the space-time and thermodynamic resources required are polynomial in L.
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"abstract": "This paper represents one approach to making explicit some of the assumptions\nand conditions implied in the widespread representation of numbers by composite\nquantum systems. Any nonempty set and associated operations is a set of natural\nnumbers or a model of arithmetic if the set and operations satisfy the axioms\nof number theory or arithmetic. This work is limited to k-ary representations\nof length L and to the axioms for arithmetic modulo k^{L}. A model of the\naxioms is described based on states in and operators on an abstract L fold\ntensor product Hilbert space H^{arith}. Unitary maps of this space onto a\nphysical parameter based product space H^{phy} are then described. Each of\nthese maps makes states in H^{phy}, and the induced operators, a model of the\naxioms. Consequences of the existence of many of these maps are discussed along\nwith the dependence of Grover\u0027s and Shor\u0027s Algorithms on these maps. The\nimportance of the main physical requirement, that the basic arithmetic\noperations are efficiently implementable, is discussed. This conditions states\nthat there exist physically realizable Hamiltonians that can implement the\nbasic arithmetic operations and that the space-time and thermodynamic resources\nrequired are polynomial in L.",
"arxiv_id": "quant-ph/0003063",
"authors": [
"Paul Benioff"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.63.032305",
"journal_ref": "Phys. Rev. A63, 032305 (2001)",
"title": "The Representation of Natural Numbers in Quantum Mechanics",
"url": "https://arxiv.org/abs/quant-ph/0003063"
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