dorsal/arxiv
View SchemaA Study of Quantum Error Correction by Geometric Algebra and Liquid-State NMR Spectroscopy
| Authors | Yehuda Sharf, David G. Cory, Shyamal S. Somaroo, Timothy F. Havel, Emanuel Knill, Raymond Laflamme |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0004030 |
| URL | https://arxiv.org/abs/quant-ph/0004030 |
| DOI | 10.1080/002689700413604 |
| Journal | Molecular Physics, 2000, Vol 98, No 17, 1347-1363 |
Abstract
Quantum error correcting codes enable the information contained in a quantum state to be protected from decoherence due to external perturbations. Applied to NMR, quantum coding does not alter normal relaxation, but rather converts the state of a ``data'' spin into multiple quantum coherences involving additional ancilla spins. These multiple quantum coherences relax at differing rates, thus permitting the original state of the data to be approximately reconstructed by mixing them together in an appropriate fashion. This paper describes the operation of a simple, three-bit quantum code in the product operator formalism, and uses geometric algebra methods to obtain the error-corrected decay curve in the presence of arbitrary correlations in the external random fields. These predictions are confirmed in both the totally correlated and uncorrelated cases by liquid-state NMR experiments on 13C-labeled alanine, using gradient-diffusion methods to implement these idealized decoherence models. Quantum error correction in weakly polarized systems requires that the ancilla spins be prepared in a pseudo-pure state relative to the data spin, which entails a loss of signal that exceeds any potential gain through error correction. Nevertheless, this study shows that quantum coding can be used to validate theoretical decoherence mechanisms, and to provide detailed information on correlations in the underlying NMR relaxation dynamics.
{
"annotation_id": "bb189296-0536-43d8-909d-4fb09d9de36c",
"date_created": "2026-03-02T18:01:38.692000Z",
"date_modified": "2026-03-02T18:01:38.692000Z",
"file_hash": "4ea58fa2cacbb1e7074fbe875ee786e813aceba41c0ac09cd933b8af129e53d6",
"private": false,
"record": {
"abstract": "Quantum error correcting codes enable the information contained in a quantum\nstate to be protected from decoherence due to external perturbations. Applied\nto NMR, quantum coding does not alter normal relaxation, but rather converts\nthe state of a ``data\u0027\u0027 spin into multiple quantum coherences involving\nadditional ancilla spins. These multiple quantum coherences relax at differing\nrates, thus permitting the original state of the data to be approximately\nreconstructed by mixing them together in an appropriate fashion. This paper\ndescribes the operation of a simple, three-bit quantum code in the product\noperator formalism, and uses geometric algebra methods to obtain the\nerror-corrected decay curve in the presence of arbitrary correlations in the\nexternal random fields. These predictions are confirmed in both the totally\ncorrelated and uncorrelated cases by liquid-state NMR experiments on\n13C-labeled alanine, using gradient-diffusion methods to implement these\nidealized decoherence models. Quantum error correction in weakly polarized\nsystems requires that the ancilla spins be prepared in a pseudo-pure state\nrelative to the data spin, which entails a loss of signal that exceeds any\npotential gain through error correction. Nevertheless, this study shows that\nquantum coding can be used to validate theoretical decoherence mechanisms, and\nto provide detailed information on correlations in the underlying NMR\nrelaxation dynamics.",
"arxiv_id": "quant-ph/0004030",
"authors": [
"Yehuda Sharf",
"David G. Cory",
"Shyamal S. Somaroo",
"Timothy F. Havel",
"Emanuel Knill",
"Raymond Laflamme"
],
"categories": [
"quant-ph"
],
"doi": "10.1080/002689700413604",
"journal_ref": "Molecular Physics, 2000, Vol 98, No 17, 1347-1363",
"title": "A Study of Quantum Error Correction by Geometric Algebra and Liquid-State NMR Spectroscopy",
"url": "https://arxiv.org/abs/quant-ph/0004030"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "218fa083-38eb-4c8c-beca-8d7e63a32eb4",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}