Transversal AND in Quantum Codes

TL;DR

This paper introduces a novel qutrit quantum error-correcting code with a transversal AND gate, enabling efficient simulation of qubit computation and advancing quantum error correction techniques.

Contribution

It constructs a new qutrit [ [6,2,2] ] code with a transversal AND gate and develops protocols for mixed qubit-qutrit codes and magic state distillation.

Findings

Constructed a [ [6,2,2] ] qutrit code with transversal AND gate.

Enhanced code distance to [ [48,2,4] ] through augmentation.

Developed protocols for qubit-qutrit mixed codes and magic state distillation.

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 $n$ 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.