A Study of Quantum Error Correction by Geometric Algebra and Liquid-State NMR Spectroscopy

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.Comment: 33 pages plus 6 figures, LaTeX article class with amsmath & graphicx package

Solid-State Dynamics in the closo-Carboranes:A (11)B MAS NMR and Molecular Dynamics Study

This work explores the dynamic behaviour of the three closo-carborane isomers (formula C2B10H12) using modern solid-state magic angle spinning (MAS) NMR techniques and relates the experimental measurements to theoretical results obtained using molecular dynamics simulations. At high temperatures and at B0 = 9.4 T, the 11B MAS linewidths are narrow (40-90 Hz) for the three isomers. The rotational correlation times (tauc) calculated by molecular dynamics are on the picosecond timescale, showing a quasi-isotropic rotation at these temperatures, typical for liquid systems. For all three isomers, the values of the 11B spin-lattice relaxation times (T1) show discontinuities as the temperature is decreased, confirming the phase changes reported in the literature. At low temperatures, the 11B MAS spectra of all three isomers exhibit much broader lines. The simulations showed that the molecular reorientation was anisotropic around different symmetry axes for each isomer, and this was supported by the values of the reduced quadrupolar parameter PQeff derived from “dynamic shift” measurements using 11B MQMAS NMR spectroscopy. The behaviour of PQeff as a function of temperature for para-carborane suggests that molecular reorientation is about the C5 symmetry axis of the molecule at low temperatures and this was supported by the molecular dynamics simulations

Chemical Shielding Anisotropies for Chloroform Exchanging between a Free Site and a Complex with Cryptophane-D: A Cross-Correlated NMR Relaxation Study

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