Transversal AND in Quantum Codes

2026-03-04

AI Summary

This work addresses the long-standing challenge of implementing a fault-tolerant AND gate in quantum error correction, which is inherently non-Clifford and irreversible on qubits. The authors propose a novel qutrit-based quantum error-correcting code that enables a transversal AND gate for the first time, constructing a [[6,2,2]] code and subsequently extending it via stabilizer enlargement and concatenation to a [[48,2,4]] code while preserving transversal logical operations. Their approach integrates the Clifford+T gate set, symmetric T-depth-1 circuits, CSS structure, and hybrid qubit–qutrit subspace encoding, complemented by a tailored magic state distillation protocol. The resulting two-qutrit AND gate achieves a T-count of only 3 and generalizes efficiently to n-qubit scenarios with T-count 3n−3, offering a promising pathway toward high-distance fault-tolerant quantum computation.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG

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.

Problem

• transversal AND gate
• quantum error-correcting code
• qutrit
• logical gate implementation
• quantum computation

Innovation

• transversal AND gate
• qutrit quantum error-correcting code
• CSS code construction
• magic state distillation
• symmetric T-depth one circuit