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arXiv:2610.27556

Hybrid Lattice Surgery between Smooth Boundaries

Grace Ding, Yanzhu Chen, Sheng-Jie Huang

quant-phcond-mat.str-elSubmitted Oct 20, 2026• Announced Oct 21, 2026• 0 citations
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Abstract

We develop smooth hybrid lattice surgery for group surface codes, enabling joint logical measurements through merge and split operations along smooth boundaries. The code patches may correspond to identical or different finite groups, including non-Abelian groups. Our central result is that smooth merge and split realize a logical $K$-coset measurement, where $K$ is the subgroup defining the interface. This generalizes the joint logical parity measurement of ordinary surface codes. Using a tensor-network representation, we derive explicit measurement circuits and develop procedures for correcting nonidentity measurement outcomes. We illustrate the framework through a generalized controlled multiplication gate between group surface codes and protocols for preparing the $T$, $CZ$, $CCX$, and $CS$ magic states, producing resources encoded in one or more $\mathbb{Z}_2$ surface code patches. Our results show how smooth hybrid lattice surgery broadens the range of logical operations available for group surface codes.

Comments
15+7 pages, 14 figures, 1 table

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We develop smooth hybrid lattice surgery for group surface codes, enabling joint logical measurements through merge and split operations along smooth boundaries. The code patches may correspond to identical or different finite groups, including non-Abelian groups. Our central result is that smooth merge and split realize a logical KKK-coset measurement, where KKK is the subgroup defining the interface. This generalizes the joint logical parity measurement of ordinary surface codes. Using a tensor-network representation, we derive explicit measurement circuits and develop procedures for correcting nonidentity measurement outcomes. We illustrate the framework through a generalized controlled multiplication gate between group surface codes and protocols for preparing the TTT, CZCZCZ, CCXCCXCCX, and CSCSCS magic states, producing resources encoded in one or more Z2\mathbb{Z}_2Z2​ surface code patches. Our results show how smooth hybrid lattice surgery broadens the range of logical operations available for group surface codes.
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.
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We introduce group surface codes, which are a natural generalization of the Z2\mathbb{Z}_2Z2​ surface code, and equivalent to quantum double models of finite groups with specific boundary conditions. We show that group surface codes can be leveraged to perform non-Clifford gates in Z2\mathbb{Z}_2Z2​ surface codes, thus enabling universal computation along with established means of performing logical Clifford gates. Moreover, for suitably chosen groups, we demonstrate that arbitrary reversible classical gates can be implemented transversally in the group surface code. We present the logical operations in terms of a set of elementary logical operations, which include transversal logical gates, a means of transferring encoded information into and out of group surface codes, and preparation and readout. By composing these elementary operations, we implement a wide variety of logical gates and provide a unified perspective on recent constructions in the literature for sliding group surface codes and preparing magic states. We furthermore use tensor networks inspired by ZX-calculus to construct spacetime implementations of the elementary operations. This spacetime perspective also allows us to establish explicit correspondences with topological gauge theories. Our work extends recent efforts in performing universal quantum computation in topological orders without the braiding of anyons, and shows how certain group surface codes allow us to bypass the restrictions set by the Bravyi-König theorem, which limits the computational power of topological Pauli stabilizer models.
Fault-tolerant quantum computation can incur a substantial resource cost when a desired logical operation must be constructed from a restricted set of logical primitives. To make such operations directly available, we introduce irrep surface codes (ISCs), two-dimensional topological quantum error-correcting codes engineered to transversally implement a prescribed set of logical gates. The target logical gates only need to satisfy two conditions: (i) they generate a finite group, which, up to global phases, is forced by the Eastin-Knill theorem, and (ii) they are irreducible, meaning that any operator commuting with the set of gates must be proportional to the identity. This accommodates arbitrary finite groups of diagonal gates and arbitrary reversible classical gates, since each can be embedded into a larger irreducible finite group of gates. It also enables logical gates outside every finite level of the Clifford hierarchy. To incorporate these operations into surface-code computations, we further develop lattice-surgery protocols to transfer the encoded information between surface codes and ISCs obeying one of two sufficient conditions. Together, these results establish a systematic method for tailoring topological codes to desired logical operations and integrating them into conventional computations.
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