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

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.

33 pages plus 6 figures, LaTeX article class with amsmath & graphicx packages

References in corpus (6)

• Experimental realization of a quantum algorithm
• Experimental Quantum Error Correction
• Implementation of a Quantum Search Algorithm on a Nuclear Magnetic Resonance Quantum Computer
• Complete quantum teleportation using nuclear magnetic resonance
• Decoherence, Einselection, and the Existential Interpretation (the Rough Guide)
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Cited by in corpus (12)

• Experimental Implementation of Remote State Preparation by Nuclear Magnetic Resonance
• Quantum Computing with NMR
• Tomographic measurements on superconducting qubit states
• Experimental Implementation of a Concatenated Quantum Error-Correcting Code
• Experimental Test of Quantum No-Hiding Theorem
• Introduction to NMR Quantum Information Processing
• Hadamard Products of Product Operators and the Design of Gradient-Diffusion Experiments for Simulating Decoherence by NMR Spectroscopy
• Geometric Algebra and Information Geometry for Quantum Computational Software
• Spatially Encoded Pseudo-Pure States for NMR Quantum Information Processing
• Suppression of decoherence in quantum registers by entanglement with a nonequilibrium environment
• Elementary gates for cartoon computation
• Quantum Codes for Controlling Coherent Evolution