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Paper 1

Improved error thresholds for measurement-free error correction

Daniel Crow, Robert Joynt, Mark Saffman

Year
2015
Journal
arXiv preprint
DOI
arXiv:1510.08359
arXiv
1510.08359

Motivated by limitations and capabilities of neutral atom qubits, we examine whether measurement-free error correction can produce practical error thresholds. We show that this can be achieved by extracting redundant syndrome information, giving our procedure extra fault tolerance and eliminating the need for ancilla verification. The procedure is particularly favorable when multi-qubit gates are available for the correction step. Simulations of the bit-flip, Bacon-Shor, and Steane codes indicate that coherent error correction can produce threshold error rates that are on the order of $10^{-3}$ to $10^{-4}$---comparable with or better than measurement-based values, and much better than previous results for other coherent error correction schemes. This indicates that coherent error correction is worthy of serious consideration for achieving protected logical qubits.

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Paper 2

$\mathbb{Z}_3$ Parafermionic Chain Emerging From Yang-Baxter Equation

Li-Wei Yu, Mo-Lin Ge

Year
2015
Journal
arXiv preprint
DOI
arXiv:1507.05269
arXiv
1507.05269

We construct the 1D $\mathbb{Z}_3$ parafermionic model based on the solution of Yang-Baxter equation and express the model by three types of fermions. It is shown that the $\mathbb{Z}_3$ parafermionic chain possesses both triple degenerate ground states and non-trivial topological winding number. Hence, the $\mathbb{Z}_3$ parafermionic model is a direct generalization of 1D $\mathbb{Z}_2$ Kitaev model. Both the $\mathbb{Z}_2$ and $\mathbb{Z}_3$ model can be obtained from Yang-Baxter equation. On the other hand, to show the algebra of parafermionic tripling intuitively, we define a new 3-body Hamiltonian $\hat{H}_{123}$ based on Yang-Baxter equation. Different from the Majorana doubling, the $\hat{H}_{123}$ holds triple degeneracy at each of energy levels. The triple degeneracy is protected by two symmetry operators of the system, $ω$-parity $P$($ω=e^{{\textrm{i}\frac{2π}{3}}}$) and emergent parafermionic operator $Γ$, which are the generalizations of parity $P_{M}$ and emergent Majorana operator in Lee-Wilczek model, respectively. Both the $\mathbb{Z}_3$ parafermionic model and $\hat{H}_{123}$ can be viewed as SU(3) models in color space. In comparison with the Majorana models for SU(2), it turns out that the SU(3) models are truly the generalization of Majorana models resultant from Yang-Baxter equation.

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