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arXiv:quant-ph/0004030 · Molecular Physics

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

Yehuda Sharf, David G. Cory, Shyamal S. Somaroo, Timothy F. Havel, Emanuel H. Knill, Raymond Laflamme, Wojciech Hubert Zurek

Published Sep 1, 2000• 26 citations

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, this procedure 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-labelled 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.

DOI: 10.1080/002689700413604

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Quantum error correcting codes enable the information contained in a quantum state to be protected from decoherence due to external perturbations. Applied to NMR, this procedure 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-labelled 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.

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Quantum error correction is required to compensate for the fragility of the state of a quantum computer. We report the first experimental implementations of quantum error correction and confirm the expected state stabilization. A precise analysis of the decay behavior is performed in alanine and a full implementation of the error correction procedure is realized in trichloroethylene. In NMR computing, however, a net improvement in the signal to noise would require very high polarization. The experiment implemented the three-bit code for phase errors using liquid state NMR.

Most quantum error-correcting codes are predicated on the assumption that there exists a reservoir of qubits in the state $|0\ensuremath{\rangle}$ which can be used as ancilla qubits to prepare multiqubit logical states. In this Brief Report, we examine the consequences of relaxing this assumption and propose a method to increase the fidelity produced by a given code when the ancilla qubits are initialized in mixed states, using the same number of qubits, at most doubling the number of gates when the recovery operation would already be implemented. The procedure implemented consists of altering the encoding operator to include the inverse of the unitary operation used to correct detected errors after decoding. This augmentation will be especially useful in quantum computing architectures that do not possess projective measurement, such as solid-state NMR quantum information processing.

Quantum computers exploit quantum mechanical phenomena as resources to process information.They are therefore far more vulnerable to errors than their classical counterparts.The main obstacle to the practical realization of a working large-scale quantum computer is decoherence, i.e., the process by which a quantum system decays to a classical state through interaction with the environment.We herein discuss two techniques to implement robust quantum information processing, and report on their implementation in a liquid state NMR Quantum Computing experiment.We describe a method to implement a two-qubit gate between spins 2 and 3 of a three-spin linear chain molecule by applying a soft pulse to spin 1, with appropriate choices of the rf-field amplitude ω 1 and duration τ [1].The Bloch-Siegert effect for such a soft pulse is negligible in general, which makes NMR pulse programming and quantum gates design much simpler than with conventional hard π-pulses; it also makes this method suitable for use with both homo-and heteronuclear molecules.We then discuss a simple Operator Quantum Error Correction (OQEC) scheme with three qubits [2].This scheme provides the simplest possible Noiseless Subsystem, in terms of the number of CNOT gates, that is robust against (restricted) fully correlated noise.We demonstrated our OQEC scheme experimentally using a three-qubit NMR Quantum Computer, in which the ancillae are in the maximally mixed state.In this experiment, we made practical use of the soft pulse technique [1] to implement the two-qubit gates in the pulse sequences realizing encoding and decoding operations.

Quantum-information processing by liquid-state NMR spectroscopy uses pseudopure states to mimic the evolution and observations on true pure states. A new method of preparing pseudopure states is described, which involves the selection of spatially labeled states of an ancilla spin with which the spin system of interest is correlated. This permits a general procedure to be given for the preparation of pseudopure states on any number of spins, subject to the limitations imposed by the loss of signal from the selected subensemble. The preparation of a single pseudopure state is demonstrated by carbon and proton NMR of

{}^{13}\mathrm{C}

-labeled alanine. With a judicious choice of magnetic-field gradients, the method further allows encoding of up to

{2}^{N}

pseudopure states in independent spatial modes in an

N+1

spin system. Fast encoding and decoding schemes are demonstrated for the preparation of four such spatially labeled pseudopure states.

Concatenated coding provides a general strategy to achieve the desired level of noise protection in quantum information processing. We report the implementation of a concatenated quantum error-correcting code able to correct phase errors with a strong correlated component. The experiment was performed using liquid-state nuclear magnetic resonance techniques on a four spin subsystem of labeled crotonic acid. Our results show that concatenation between active and passive quantum error correction is a practical tool to handle realistic noise involving both independent and correlated errors.