paper

Transversal AND in Quantum Codes

arXiv:2603.04548 · doi:10.1007/978-3-032-30839-9_1

Abstract

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 qubits with T-count . Our main result is the construction of a novel qutrit 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 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.

40 pages

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