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Paper 1

Transversal AND in Quantum Codes

Christine Li, Lia Yeh

Year

2026

Journal

arXiv preprint

DOI

arXiv:2603.04548

arXiv

2603.04548

The AND gate is not reversible$\unicode{x2014}$on qubits. However, it is reversible on qutrits, making it a building block for efficient simulation of qubit computation using qutrits. We first observe that there are multiple two-qutrit Clifford+T unitaries that realize the AND gate with T-count 3, and its generalizations to $n$ qubits with T-count $3n-3$. Our main result is the construction of a novel qutrit $\mathopen{[\![} 6,2,2 \mathclose{]\!]}$ quantum error-correcting code with a transversal implementation of the AND gate. The key insight in our approach is that a symmetric T-depth one circuit decomposition$\unicode{x2014}$composed of a CX circuit, T and T dagger gates, followed by the CX circuit in reverse$\unicode{x2014}$of a given unitary can be interpreted as a CSS code. We can increase the code distance by augmenting the code circuit with additional stabilizers while preserving the logical gate. This results in a code with a "built-in" transversal implementation of the original unitary, which can be further concatenated to attain a $\mathopen{[\![} 48,2,4 \mathclose{]\!]}$ code with the same transversal logical gate. Furthermore, we present several protocols for mixed qubit-qutrit codes which we call Qubit Subspace Codes, and for magic state distillation and injection.

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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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