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

Demonstrating Quantum Error Correction that Extends the Lifetime of Quantum Information

The remarkable discovery of Quantum Error Correction (QEC), which can overcome the errors experienced by a bit of quantum information (qubit), was a critical advance that gives hope for eventually realizing practical quantum computers. In principle, a system that implements QEC can actually pass a "break-even" point and preserve quantum information for longer than the lifetime of its constituent parts. Reaching the break-even point, however, has thus far remained an outstanding and challenging goal. Several previous works have demonstrated elements of QEC in NMR, ions, nitrogen vacancy (NV) centers, photons, and superconducting transmons. However, these works primarily illustrate the signatures or scaling properties of QEC codes rather than test the capacity of the system to extend the lifetime of quantum information over time. Here we demonstrate a QEC system that reaches the break-even point by suppressing the natural errors due to energy loss for a qubit logically encoded in superpositions of coherent states, or cat states of a superconducting resonator. Moreover, the experiment implements a full QEC protocol by using real-time feedback to encode, monitor naturally occurring errors, decode, and correct. As measured by full process tomography, the enhanced lifetime of the encoded information is 320 microseconds without any post-selection. This is 20 times greater than that of the system's transmon, over twice as long as an uncorrected logical encoding, and 10% longer than the highest quality element of the system (the resonator's 0, 1 Fock states). Our results illustrate the power of novel, hardware efficient qubit encodings over traditional QEC schemes. Furthermore, they advance the field of experimental error correction from confirming the basic concepts to exploring the metrics that drive system performance and the challenges in implementing a fault-tolerant system

Iterative Soft Input Soft Output Decoding of Reed-Solomon Codes by Adapting the Parity Check Matrix

An iterative algorithm is presented for soft-input-soft-output (SISO) decoding of Reed-Solomon (RS) codes. The proposed iterative algorithm uses the sum product algorithm (SPA) in conjunction with a binary parity check matrix of the RS code. The novelty is in reducing a submatrix of the binary parity check matrix that corresponds to less reliable bits to a sparse nature before the SPA is applied at each iteration. The proposed algorithm can be geometrically interpreted as a two-stage gradient descent with an adaptive potential function. This adaptive procedure is crucial to the convergence behavior of the gradient descent algorithm and, therefore, significantly improves the performance. Simulation results show that the proposed decoding algorithm and its variations provide significant gain over hard decision decoding (HDD) and compare favorably with other popular soft decision decoding methods.Comment: 10 pages, 10 figures, final version accepted by IEEE Trans. on Information Theor