A study of quantum error correction by geometric algebra and liquid-state NMR spectroscopy

Quantum error correcting codes enable the information contained in a quantu m state to be protected from decoherence due to external perturbations. App lied to NMR, this procedure does not alter normal relaxation, but rather co nverts the state of a 'data' spin into multiple quantum coherences involvin g additional ancilla spins. These multiple quantum coherences relax at diff ering rates, thus permitting the original state of the data to be approxima tely reconstructed by mixing them together in an appropriate fashion. This paper describes the operation of a simple, three-bit quantum code in the pr oduct operator formalism, and uses geometric algebra methods to obtain the error-corrected decay curve in the presence of arbitrary correlations in th e external random fields. These predictions are confirmed in both the total ly correlated and uncorrelated cases by liquid-state NMR experiments on C-1 3-labelled alanine, using gradient- diffusion methods to implement these id ealized decoherence models. Quantum error correction in weakly polarized sy stems requires that the ancilla spins be prepared in a pseudo-pure state re lative to the data spin, which entails a loss of signal that exceeds any po tential 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 r elaxation dynamics.