dorsal/arxiv
View SchemaKirchhoff's Rule for Quantum Wires. II: The Inverse Problem with Possible Applications to Quantum Computers
| Authors | Vadim Kostrykin, Robert Schrader |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9910053 |
| URL | https://arxiv.org/abs/quant-ph/9910053 |
| DOI | 10.1002/1521-3978(200008)48:8<703::AID-PROP703>3.0.CO;2-O |
| Journal | Fortschritte der Physik 48 (2000), 703 - 716 |
Abstract
In this article we continue our investigations of one particle quantum scattering theory for Schroedinger operators on a set of connected (idealized one-dimensional) wires forming a graph with an arbitrary number of open ends. The Hamiltonian is given as minus the Laplace operator with suitable linear boundary conditions at the vertices (the local Kirchhoff law). In ``Kirchhoff's rule for quantum wires'' [J. Phys. A: Math. Gen. 32, 595 - 630 (1999)] we provided an explicit algebraic expression for the resulting (on-shell) S-matrix in terms of the boundary conditions and the lengths of the internal lines and we also proved its unitarity. Here we address the inverse problem in the simplest context with one vertex only but with an arbitrary number of open ends. We provide an explicit formula for the boundary conditions in terms of the S-matrix at a fixed, prescribed energy. We show that any unitary $n\times n$ matrix may be realized as the S-matrix at a given energy by choosing appropriate (unique) boundary conditions. This might possibly be used for the design of elementary gates in quantum computing. As an illustration we calculate the boundary conditions associated to the unitary operators of some elementary gates for quantum computers and raise the issue whether in general the unitary operators associated to quantum gates should rather be viewed as scattering operators instead of time evolution operators for a given time associated to a quantum mechanical Hamiltonian.
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"abstract": "In this article we continue our investigations of one particle quantum\nscattering theory for Schroedinger operators on a set of connected (idealized\none-dimensional) wires forming a graph with an arbitrary number of open ends.\nThe Hamiltonian is given as minus the Laplace operator with suitable linear\nboundary conditions at the vertices (the local Kirchhoff law). In ``Kirchhoff\u0027s\nrule for quantum wires\u0027\u0027 [J. Phys. A: Math. Gen. 32, 595 - 630 (1999)] we\nprovided an explicit algebraic expression for the resulting (on-shell) S-matrix\nin terms of the boundary conditions and the lengths of the internal lines and\nwe also proved its unitarity. Here we address the inverse problem in the\nsimplest context with one vertex only but with an arbitrary number of open\nends. We provide an explicit formula for the boundary conditions in terms of\nthe S-matrix at a fixed, prescribed energy. We show that any unitary $n\\times\nn$ matrix may be realized as the S-matrix at a given energy by choosing\nappropriate (unique) boundary conditions. This might possibly be used for the\ndesign of elementary gates in quantum computing. As an illustration we\ncalculate the boundary conditions associated to the unitary operators of some\nelementary gates for quantum computers and raise the issue whether in general\nthe unitary operators associated to quantum gates should rather be viewed as\nscattering operators instead of time evolution operators for a given time\nassociated to a quantum mechanical Hamiltonian.",
"arxiv_id": "quant-ph/9910053",
"authors": [
"Vadim Kostrykin",
"Robert Schrader"
],
"categories": [
"quant-ph"
],
"doi": "10.1002/1521-3978(200008)48:8\u003c703::AID-PROP703\u003e3.0.CO;2-O",
"journal_ref": "Fortschritte der Physik 48 (2000), 703 - 716",
"title": "Kirchhoff\u0027s Rule for Quantum Wires. II: The Inverse Problem with Possible Applications to Quantum Computers",
"url": "https://arxiv.org/abs/quant-ph/9910053"
},
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