[论文解读] Transversal AND in Quantum Codes

Christine Li, Lia Yeh|arXiv (Cornell University)|Mar 4, 2026

Quantum Computing Algorithms and Architecture被引用 0

一句话总结

作者构建了一个 qutrit CSS 代码 J6,2,2K，具备横向传输 AND 门；展示通过级联可将代码距离提升至 J48,2,4K，并引入混合比特-三态（qubit-qutrit）编码与魔态协议。

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.

研究动机与目标

• Motivate the use of qutrits to realize a reversible AND gate within fault-tolerant quantum computation.
• Construct a qutrit stabilizer code around a transversal binary AND operation.
• Demonstrate how to raise code distance via code concatenation while preserving the transversal gate.
• Introduce mixed-dimensional (qubit-qutrit) code notions and protocols for magic state distillation and injection.

提出的方法

• Leverage exact circuit synthesis to relate symmetric T-depth one circuits to CSS encoders, enabling transversal non-Clifford operations.
• Use ZX-calculus to reinterpret and extract stabilizers and logical operators from phase-gadget based encoders.
• Derive a symmetric, T-depth 1 circuit that implements the |0⟩-controlled Z gate and identify its CSS structure.
• Read off X-type and Z-type stabilizers and logical operators from the encoder forms to obtain the J6,2,2K code implementing transversal AND.
• Concatenate inner J6,2,2K with an outer code (J8,1,2K) to achieve distance 4, yielding J48,2,4K with transversal AND.
• Extend the framework to Qubit Subspace Codes and provide protocols for magic-state distillation and deterministic injection.

实验结果

研究问题

• RQ1 How can one construct a non-trivial qutrit QEC code that admits a transversal implementation of a qubit logical gate (specifically AND)?
• RQ2 Can the code distance be increased while preserving the transversal logical gate, and how does concatenation achieve this?
• RQ3 What are the benefits and construction methods for mixed qubit-qutrit codes (Qubit Subspace Codes) and how can they support fault-tolerant operations and magic-state protocols?

主要发现

• A binary classical logic gate set (AND/OR/NOT) can be emulated exactly in the qutrit Clifford+T set with T-count 3 for the 2-qutrit AND construction.
• A T-depth 1 symmetric circuit for the qubit AND enables a J6,2,2K qutrit code with transversal AND.
• The derived code has distance 2; by adding an X-type stabilizer and using ZX-calculus rewrites, the authors obtain a symmetric encoder for the transversal AND.
• Code concatenation with an outer J8,1,2K code yields a distance-4 J48,2,4K code implementing transversal AND (24 T and 24 T† gates).
• The work introduces Qubit Subspace Codes, showing projections that map logical qutrits into the qubit subspace, enabling mixed-dimension coding strategies.
• The paper discusses deterministic injection and distillation of the |0⟩-controlled Z/K gate (AND) as part of magic-state protocols for qutrit CSS codes.