dorsal/arxiv
View SchemaThe effective Hamiltonian of the Pound-Overhauser controlled-NOT gate
| Authors | D. G. Cory, A. E. Dunlop, T. F. Havel, S. S. Somaroo, W. Zhang |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9809045 |
| URL | https://arxiv.org/abs/quant-ph/9809045 |
| Journal | Concepts Magn.Reson.Part A 23:49-62,2004 |
Abstract
In NMR-based quantum computing, it is known that the controlled-NOT gate can be implemented by applying a low-power, monochromatic radio-frequency field to one peak of a doublet in a weakly-coupled two-spin system. This is known in NMR spectroscopy as Pound-Overhauser double resonance. The ``transition'' Hamiltonian that has been associated with this procedure is however only an approximation, which ignores off-resonance effects and does not correctly predict the associated phase factors. In this paper, the exact effective Hamiltonian for evolution of the spins' state in a rotating frame is derived, both under irradiation of a single peak (on-transition) as well as between the peaks of the doublet (on-resonance). The accuracy of these effective Hamiltonians is validated by comparing the observable product operator components of the density matrix obtained by simulation to those obtained by fitting the corresponding experiments. It is further shown how both the on-transition and on-resonance fields can be used to implement the controlled-NOT gate exactly up to conditional phases, and analytic expressions for these phases are derived. In Appendices, the on-resonance Hamiltonian is analytically diagonalized, and proofs are given that, in the weak-coupling approximation, off-resonance effects can be neglected whenever the radio-frequency field power is small compared to the difference in resonance frequencies of the two spins.
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"abstract": "In NMR-based quantum computing, it is known that the controlled-NOT gate can\nbe implemented by applying a low-power, monochromatic radio-frequency field to\none peak of a doublet in a weakly-coupled two-spin system. This is known in NMR\nspectroscopy as Pound-Overhauser double resonance. The ``transition\u0027\u0027\nHamiltonian that has been associated with this procedure is however only an\napproximation, which ignores off-resonance effects and does not correctly\npredict the associated phase factors. In this paper, the exact effective\nHamiltonian for evolution of the spins\u0027 state in a rotating frame is derived,\nboth under irradiation of a single peak (on-transition) as well as between the\npeaks of the doublet (on-resonance). The accuracy of these effective\nHamiltonians is validated by comparing the observable product operator\ncomponents of the density matrix obtained by simulation to those obtained by\nfitting the corresponding experiments. It is further shown how both the\non-transition and on-resonance fields can be used to implement the\ncontrolled-NOT gate exactly up to conditional phases, and analytic expressions\nfor these phases are derived. In Appendices, the on-resonance Hamiltonian is\nanalytically diagonalized, and proofs are given that, in the weak-coupling\napproximation, off-resonance effects can be neglected whenever the\nradio-frequency field power is small compared to the difference in resonance\nfrequencies of the two spins.",
"arxiv_id": "quant-ph/9809045",
"authors": [
"D. G. Cory",
"A. E. Dunlop",
"T. F. Havel",
"S. S. Somaroo",
"W. Zhang"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"journal_ref": "Concepts Magn.Reson.Part A 23:49-62,2004",
"title": "The effective Hamiltonian of the Pound-Overhauser controlled-NOT gate",
"url": "https://arxiv.org/abs/quant-ph/9809045"
},
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