Transversal Toffoli-gate in Hybrid-code System

TL;DR

The paper tackles the challenge of achieving a universal fault-tolerant gate set by exploring transversal Toffoli operations within a hybrid-code framework that pairs a triorthogonal CSS code with its mirrored code. It proves a fully transversal Toffoli-gate is possible across three code blocks and introduces a layer-by-layer Toffoli-state distillation that obviates the need for pre-distilled $\mathbf{T}$ gates. The authors provide a detailed decomposition and circuit-level analysis showing transversal CZ/CNOT between the paired codes and propose a practical Toffoli-state distillation protocol with significant resource savings, supported by numerical simulations. The work offers a concrete route to universal fault-tolerant computation with reduced qubit overhead and highlights future directions to generalize transversal gates across multiple codes.

Abstract

We study the transversality of the Toffoli gate in a hybrid-code system that employs two quantum error correction codes with special structure. We find that a system using a triorthogonal code with its paired code supports a fully transversal implementation of the Toffoli gate. Through circuit-level analysis, we prove the transversality of the Toffoli operation in this system. Based on this hybrid-code framework, we propose a Toffoli state distillation protocol that does not rely on pre-distilled $\mathbf{T}$-gate magic states. In our approach, the Toffoli state is directly distilled layer by layer within the hybrid-code system using only transversal operations. Numerical simulations demonstrate that our method uses approximately 50\% fewer qubit resources than previously reported protocols.

Figures (6)

• Figure 1: A Toffoli-gate is equivalent to a CCZ gate with target qubit conjugated by Hadamard gates.
• Figure 2: Decomposition of a Toffoli-gate into 6 CNOT gates, 2 Hadamard gates and 7 $\mathbf{T}$ (or $\mathbf{T}^\dagger$) gates.
• Figure 3: Modified Toffoli-gate decomposition circuit. The first and second qubits encoded in $\mathcal{Q}^{T}$ and the last one encoded in $\mathcal{Q}^{T_X}$.
• Figure 4: Circuit for both Toffoli state distillation and Toffoli gate implementation. When the Toffoli state is used to perform a Toffoli gate, $\mathcal{Q}^1$ and $\mathcal{Q}^2$ correspond to the same code employed in the computational system, such as the surface code. In the Toffoli-state distillation process, the $\mathcal{Q}^1$ and $\mathcal{Q}^2$ are chosen as $\mathcal{Q}^{\rm T}$ and $\mathcal{Q}^{\rm T_x}$ respectively. The orange dashed box outputs an ancillary Toffoli-state. Using CNOTs and measurements, the Toffoli-gate action is teleported to the 3-qubit state $\lvert a\rangle\lvert b\rangle\lvert c\rangle$.
• Figure 5: Comparison of qubit resource costs for different Toffoli state distillation protocols. The 15-to-1 and $3k+8$-to-$k$ Toffoli curves corresponding to our direct Toffoli state distillation protocols using the $[[15,1,3]]$ and $[[3k+8,k,2]]$ codes, respectively. The corresponding 'Magic' curves represent the protocols that first distill seven magic states using the same codes to construct a Toffoli state. With the initial physical error rate selected as $10^{-2}$, our direct Toffoli-state distillation approach achieves the same target error rate with approximately $50\%$ fewer qubits.

Theorems & Definitions (6)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof