Logical gates on Floquet codes via folds and twists

TL;DR

Floquet codes protect quantum information with time-dependent stabilisers, but implementing fault-tolerant logical gates is nontrivial. This work adapts two static-code techniques—fold-transversal gates and Dehn twists—to Floquet codes, demonstrating Hadamard-type and S-type gates via ZX-duality and CNOTs via lattice distortions, respectively. Benchmarking on the CSS honeycomb Floquet code reveals a fault-tolerance threshold around 0.25–0.35% with sub-threshold exponential error suppression, and embedded-code discussions illuminate alternative pathways to gates. The results show robust, near-memory-like gate performance, while highlighting hardware-connectivity needs and decoding considerations for non-graph-like errors, and point to extensions to other lattices and higher-rate Floquet codes.

Abstract

Floquet codes have recently emerged as a new family of error-correcting codes, and have drawn significant interest across both theoretical and practical quantum computing. A central open question has been how to implement logical operations on these codes. In this work, we show how two techniques from static quantum error-correcting codes can also be implemented on Floquet codes. First, we present a way of implementing fold-transversal operations on Floquet codes in order to yield logical Hadamard and S gates. And second, we present a way of implementing logical CNOT gates on Floquet codes via Dehn twists. We discuss the requirements for these techniques, and show that they are applicable to a wide family of Floquet codes defined on colour code lattices. Through numerical benchmarking of the logical operations on the CCS Floquet code, we establish a logical-gate threshold of 0.25-0.35% and verify sub-threshold exponential error suppression. Our results show that these logical operations are robust, featuring a performance that is close to the baseline set by a quantum memory benchmark. Finally, we explain in detail how to implement logical gates on Floquet codes by operating on the embedded codes.

Figures (40)

• Figure 1: Logical-gate techniques adapted and applied to Floquet codes. (a) In \ref{['sec:fold-transversal-floquet']}, we extend the concept of fold-transversal logical gates to Floquet codes, and show how this can be utilised to implement logical Hadamard and $\operatorname{S}$ gates. (b) In \ref{['sec:dehn-twist-floquet']}, we show how distortion of the Floquet code lattice can be used to implement $\operatorname{CNOT}$ gates via Dehn twists. (c) In \ref{['sec:embedded-codes']}, we investigate how to implement logical gates on the embedded stabiliser codes.
• Figure 2: Fold-transversal gates on the unrotated toric code. An example of unrotated toric code along with logical operators is shown in (a). Nodes represent physical qubits, and red & blue plaquettes denote $\operatorname{X}$ & $\operatorname{Z}$ stabilisers, respectively. The fold line for the $\operatorname{ZX}$-duality is shown as a dashed line. (b) A fold-transversal Hadamard-type gate $\operatorname{H_0}\otimes \operatorname{H_1}$ is implemented by applying transversal Hadamard gates to every qubit and $\operatorname{SWAP}$ gates between pairs of qubits across the fold line. (c) A fold-transversal $\operatorname{S}$-type gate $\operatorname{S_0}\otimes \operatorname{S_1}$ is implemented by applying $\operatorname{S}$ gates along the fold line and $\operatorname{CZ}$ gates between pairs of qubits across the fold line.
• Figure 3: Dehn twist schematic on a torus. In (a), independent non-trivial loops along which the logical operators lie are shown. The Dehn twist of the toroidal loop (red) along the poloidal loop is shown in (b). The Dehn twist along the toroidal loop is displayed in (c).
• Figure 4: Linear Dehn twist implementing logical $\operatorname{CNOT}_{0,1}$. The selection of logical operators and the initial layer of $\operatorname{CNOT}$ gates are illustrated in (a). Note that the support of $\operatorname{Z}_0$ and $\operatorname{X}_1$ remains the same in subsequent figures, hence omitted. The deformed lattice, modifications to logical operators, and the $\operatorname{CNOT}$ sequence for subsequent layers are presented in (b) and (c). After the application of $\operatorname{CNOT}$ gates in (c), the original lattice structure is restored in (d), and operators $\operatorname{X}_0$ and $\operatorname{Z}_1$ acquire a component along the vertical loop.
• Figure 5: Instantaneous Dehn twist implementing logical $\operatorname{CNOT}_{0,1}$. The $\operatorname{CNOT}$ sequence along with logicals is indicated in (a). The support of $\operatorname{Z}_0$ and $\operatorname{X}_1$ remains the same throughout the process. The transformed stabilisers and logicals are presented in (b). An indicative cyclic shift is also shown in (b). Note that the range increases from left to right. When the lattice is restored in (c), the operators $\operatorname{X}_0$ and $\operatorname{Z}_1$ gain the vertical component, thus finishing the protocol.