Hybrid Lattice Surgery: Non-Clifford Gates via Non-Abelian Surface Codes

TL;DR

This work introduces hybrid lattice surgery that couples Abelian surface-code patches with non-Abelian quantum doubles $D(G)$ to realize logical non-Clifford gates more efficiently. By performing carefully designed rough merges and splits across interfaces characterized by diagonal subgroups and condensable algebras, the authors implement magic-state generation and gate teleportation within a 2D architecture, while providing a complementary continuum TQFT description of the interfaces. The construction centers on concrete protocols involving $D( ext{Z}_4)$, $D(D_4)$, and $D( ext{Z}_2 imes ext{Z}_2)$ (and extensions via $D(S_3)$), achieving $T$ and $T^{1/n}$-type operations and enabling magic states for qubits and qutrits. They also develop fault-tolerance considerations, including non-Abelian syndrome handling with just-in-time decoding, and discuss generalizations to other groups and higher levels of the Clifford hierarchy. Overall, the framework offers a resource-efficient, locality-preserving route to universal quantum computing within 2D hardware by leveraging hybrid code patches and topological interfaces.

Abstract

In universal fault-tolerant quantum computing, implementing logical non-Clifford gates often demands substantial spacetime resources for many error-correcting codes, including the high-threshold surface code. A critical mission for realizing large-scale quantum computing is to develop simple and resource-efficient implementations of logical non-Clifford gates. We propose a novel way of implementing non-Clifford operations in the standard surface code based on hybrid lattice surgery. First we generalize the standard lattice surgery to hybrid lattice surgery, where operations of rough merge and rough split happen across different topological codes. Then we apply such procedures between Abelian and non-Abelian codes and show that this can provide non-Clifford operations in the standard surface code, in the form of a magic state or a non-Clifford gate teleportation. Complementing this, we provide a continuum topological field theory description of this hybrid lattice surgery utilizing interfaces between (2+1)d topological orders. From these considerations, we can generalize our protocol to non-Clifford gates and magic states at all finite levels of the Clifford hierarchy, as well as gates beyond the hierarchy. We also discuss protocols extending this framework to qutrits.

Figures (13)

• Figure 1: (a) Code patches for two surface codes realizing the quantum doubles with groups $G$ and $G'$. In both surface codes, we choose the boundary conditions to be rough on the vertical edges and smooth on the horizontal. (b) A hybrid code patch after a rough merge between them, resulting in an interface specified by $\mathcal{A}$ (a particular algebra in the quantum double).
• Figure 2: Illustration of the lattice surgery operations required for implementing a non-Clifford operation (either $T$ magic state or gate teleportation) on the standard surface code, with time flowing from bottom to top. In practice, the time represents the QEC rounds. The three code patches involved are encoded in the quantum double surface codes associated with groups ${\mathbb Z}_4, D_4, {\mathbb Z}_2$, respectively, where the one with ${\mathbb Z}_2$ is the standard surface code. The protocol starts by initializing a stabilizer state in the $D(\mathbb{Z}_4)$ code patch and performing logical measurements via a sequence of rough merge and split operations, followed by additional measurements in the $D(D_4)$ code patch. The two merge-split processes on the left and right can happen simultaneously. We omit the other measurements or transversal gates in the protocol in this diagram.
• Figure 3: (a) A boundary site $(v,p)$. (b) The boundary conditions for a $D(G)$ code patch, given by the labels of the anyons that can end on the boundaries. $R$ is an irreducible representation (irrep) of $G$ and labels an electric anyon while $[g]$ is a conjugacy class and labels a magnetic anyon.
• Figure 4: A logical operator labeled by $g$ along the left (a) or the right (b) boundary.
• Figure 5: The interface between two code patches $D(G)$ and $D(G^\prime)$ viewed as a boundary of a $D(G\times G^\prime)$ patch. In the lower half of the folded patch (the left hand side), the edge orientations are reversed compared to the patch with an interface (the right hand side) and we denote the half patch with $\overline{D(G^\prime)}$.