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Paper 1
Transversal Clifford-Hierarchy Gates via Non-Abelian Surface Codes
Alison Warman, Sakura Schafer-Nameki
- Year
- 2025
- Journal
- arXiv preprint
- DOI
- arXiv:2512.13777
- arXiv
- 2512.13777
We present an entirely 2D transversal realization of phase gates at any level of the Clifford hierarchy, and beyond, using non-Abelian surface codes. Our construction encodes a logical qubit in the quantum double $D(G)$ of a non-Abelian group $G$ on a triangular spatial patch. The logical gate is implemented transversally by stacking on the spatial region a symmetry-protected topological (SPT) phase specified by a group 2-cocycle. The Bravyi--König theorem limits the unitary gates implementable by constant-depth quantum circuits on Pauli stabilizer codes in $D$ dimensions to the $D$-th level of the Clifford hierarchy. We bypass this limitation, by constructing transversal unitary gates at arbitrary levels of the Clifford hierarchy purely in 2D, without sacrificing locality or fault tolerance, at the cost of using the quantum double of a non-Abelian group $G$. Specifically, for $G = D_{4N}$, the dihedral group of order $8N$, we realize the phase gate $T^{1/N} = \mathrm{diag}(1, e^{iπ/(4N)})$ in the logical $\overline{Z}$ basis. In this context we propose a non-abelian stabilizer group formalism, which we work out for dihedral groups. For $8N = 2^n$, the logical gate lies at the $n$-th level of the Clifford hierarchy and, importantly, has a qubit-only realization: we show that it can be constructed in terms of Clifford-hierarchy stabilizers for a code with $n$ physical qubits on each edge of the lattice. We also discuss code-switching to the $\mathbb{Z}_2 \times \mathbb{Z}_2$ and $\mathbb{Z}_2$ surface-codes, which can be utilized for the quantum error correction in this setup.
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Proceedings 9th Workshop on Quantum Physics and Logic
Ross Duncan, Prakash Panangaden
- Year
- 2014
- Journal
- arXiv preprint
- DOI
- arXiv:1407.8427
- arXiv
- 1407.8427
This volume contains the proceedings of the ninth workshop on Quantum Physics and Logic (QPL2012) which took place in Brussels from the 10th to the 12th of October 2012. QPL2012 brought together researchers working on mathematical foundations of quantum physics, quantum computing, and spatio-temporal causal structures. The particular focus was on the use of logical tools, ordered algebraic and category-theoretic structures, formal languages, semantical techniques, and other computer science methods for the study of physical behaviour in general.
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