Transversal Toffoli-gate in Hybrid-code System

TL;DR

This paper demonstrates a fully transversal Toffoli gate implementation in a hybrid quantum error correction system using triorthogonal codes, enabling resource-efficient Toffoli state distillation without pre-distilled magic states.

Contribution

The authors introduce a hybrid-code framework supporting transversal Toffoli gates and propose a resource-efficient Toffoli state distillation protocol that reduces qubit usage.

Findings

Supports a fully transversal Toffoli gate in hybrid codes

Proposes a Toffoli state distillation method without pre-distilled magic states

Achieves approximately 50% fewer qubits compared to previous protocols

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.