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Universal quantum computation with group surface codes
Quantum Physics · Submitted 5 Mar 2026
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that group surface codes can be leveraged to perform non-Clifford gates in $\mathbb{Z}_2$ surface codes, thus enabling universal computation along with established means of performing logical Clifford gates.
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The abstract, sentence by sentence
We introduce group surface codes, which are a natural generalization of the $\mathbb{Z}_2$ 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 $\mathbb{Z}_2$ 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-K\"onig theorem, which limits the computational power of topological Pauli stabilizer models.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We introduce group surface codes, which are a natural generalization of the $\mathbb{Z}_2$ surface code, and equivalent to quantum double models of finite groups with specific boundary conditions. | contribution 0.99 | 0.76 | 0.08 | asserted 1.00 |
| s1 | We show that group surface codes can be leveraged to perform non-Clifford gates in $\mathbb{Z}_2$ surface codes, thus enabling universal computation along with established means of performing logical Clifford gates. | result 0.73 | 0.92 | 0.68 | asserted 1.00 |
| s2 | Moreover, for suitably chosen groups, we demonstrate that arbitrary reversible classical gates can be implemented transversally in the group surface code. | result 0.78 | 0.93 | 0.12 | asserted 0.99 |
| s3 | 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. | contribution 0.51 | 0.92 | 0.08 | asserted 1.00 |
| s4 | 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. | contribution 0.71 | 0.93 | 0.11 | asserted 1.00 |
| s5 | We furthermore use tensor networks inspired by ZX-calculus to construct spacetime implementations of the elementary operations. | method 0.91 | 0.94 | 0.04 | asserted 1.00 |
| s6 | This spacetime perspective also allows us to establish explicit correspondences with topological gauge theories. | interpretation 0.51 | 0.92 | 0.06 | asserted 0.99 |
| s7 | 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-K\"onig theorem, which limits the computational power of topological Pauli stabilizer models. | interpretation 0.81 | 0.88 | 0.51 | asserted 1.00 |
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The paper
Original abstract
We introduce group surface codes, which are a natural generalization of the $\mathbb{Z}_2$ 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 $\mathbb{Z}_2$ 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-K\"onig theorem, which limits the computational power of topological Pauli stabilizer models.
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References
How the literature uses it
5 arXiv papers cite this in 7 sentences
Papers citing it · 5 sentences without a confident label
Cites it as background 2 papers
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Parametrised constructions of quantum Low Density Parity Check (qLDPC) Error Correcting Codes (ECC) draw from several mathematical constructs like polynomials [ref] and graphs [ref] , but ultimately require an assignment of these parameters for practical use, and [ref] raises the question of finding good parameters.
Designing Quantum Error Correcting Codes to fit decoders via Reinforcement Learning · Related Work · 0.89
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Since then, the Clifford hierarchy has been studied extensively, and has found numerous applications in quantum computation and the design of fault-tolerant protocols [ref] .
On Clifford hierarchy testing and near-extremizers of noncommutative uniformity norms · Introduction · 0.93
What it cites · 35 sentences without a confident label
Builds on 4 papers
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The corresponding spacetime logical blocks are shown below: [equation] The protocol involves the following basic steps: * Initialize a $_4^r$-GSC in the state [equation] which was also the starting point of Ref. [ref] .
Generating logical magic states with the aid of non-Abelian topological order · Preparing a $T$ state · 0.86
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These constructions generalize the magic-state preparation in Refs. [ref] and the protocols for sliding in Refs. [ref] .
Universal fault tolerant quantum computation in 2D without getting tied in knots · Introduction · 1.00
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These constructions generalize the magic-state preparation in Refs. [ref] and the protocols for sliding in Refs. [ref] .
Non-Clifford gates between stabilizer codes via non-Abelian topological order · Introduction · 1.00
Compares with 4 papers
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This differs from the sliding in Ref. [ref] by rotating the $_2^b$-GSC by an angle 90 degrees.[The boundary conditions of the $D_4$-GSC also need to be appropriately modified due to the rotated $_2^b$-GSC.] On the other hand, the sliding described in Ref. [ref] can be obtained by taking $H=_3$, $K=_2$, and $G=H ⋊K$.
Non-Clifford gates between stabilizer codes via non-Abelian topological order · Sliding · 0.95
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This differs from the sliding in Ref. [ref] by rotating the $_2^b$-GSC by an angle 90 degrees.[The boundary conditions of the $D_4$-GSC also need to be appropriately modified due to the rotated $_2^b$-GSC.] On the other hand, the sliding described in Ref. [ref] can be obtained by taking $H=_3$, $K=_2$, and $G=H ⋊K$.
Universal fault tolerant quantum computation in 2D without getting tied in knots · Sliding · 0.95
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Ref. [ref] considers untwisted quantum double models but implements logical gates associated to group cohomology.
Constant-Depth Clifford-Hierarchy Gates via Non-Abelian Surface Codes · Introduction · 0.94
Cites as background 10 papers
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Ref. [ref] previously showed how to realize a logical $CCZ$ gate on three surface codes by sliding two of the patches across the third, with the intermediate code realizing a twisted $_2^3$ quantum double (topologically equivalent to the $D_4$ quantum double).
Universal fault tolerant quantum computation in 2D without getting tied in knots · Sliding · 0.84
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It has long been appreciated that non-Abelian anyons can be used to store and process quantum information through braiding and fusion [ref] –in some cases, allowing for a universal set of quantum gates [ref] .
A modular functor which is universal for quantum computation · Introduction · 0.94
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Recent works have demonstrated how non-Abelian topological codes can be used to prepare a variety of magic states [ref] .
Generating logical magic states with the aid of non-Abelian topological order · Magic state preparation · 0.88
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