We show how to perform scalable fault-tolerant non-Clifford gates in two dimensions by introducing domain walls between the surface code and a non-Abelian topological code whose code space is stabilized by Clifford operators. We formulate a path-integral framework which provides both a macroscopic picture for different logical gates as well as a way to derive the associated microscopic circuits. We also show an equivalence between our approach and prior proposals where a 2D array of qubits reproduces the action of a transversal gate in a 3D stabilizer code over time, thus establishing a connection between 3D codes and 2D non-Abelian topological phases. We prove a threshold theorem for our protocols under local stochastic circuit noise using a just-in-time decoder to correct the non-Abelian code.
", "doi": "10.1103/vrty-qs5h", "issn": "2160-3308", "publisher": "American Physical Society", "publication": "Physical Review X", "publication_date": "2026-07-07", "series_number": "3", "volume": "16", "issue": "3", "pages": "031001" }, { "id": "authors:9qy2n-vnc51", "collection": "authors", "collection_id": "9qy2n-vnc51", "cite_using_url": "https://authors.library.caltech.edu/records/9qy2n-vnc51", "type": "article", "title": "Cups and Gates I: Cohomology Invariants and Logical\u00a0Quantum Operations", "author": [ { "family_name": "Breuckmann", "given_name": "Nikolas P.", "orcid": "0000-0002-7211-5515" }, { "family_name": "Davydova", "given_name": "Margarita", "orcid": "0000-0001-8482-5832", "clpid": "Davydova-Margarita" }, { "family_name": "Eberhardt", "given_name": "Jens N.", "orcid": "0000-0003-0577-9159" }, { "family_name": "Tantivasadakarn", "given_name": "Nathanan", "orcid": "0000-0001-5295-2124", "clpid": "Tantivasadakarn-Nathanan" } ], "abstract": "We take initial steps towards a general framework for constructing logical gates in general quantum CSS codes. Viewing CSS codes as cochain complexes, we observe that cohomology invariants naturally give rise to diagonal logical gates. We show that such invariants exist if the quantum code has a structure that relaxes certain properties of a differential graded algebra. We show how to equip quantum codes with such a structure by defining cup products on CSS codes. The logical gates obtained from this approach can be implemented by a constant-depth unitary circuit. In particular, we construct a Λ-fold cup product that can produce a logical operator in the Λ-th level of the Clifford hierarchy on Λ copies of the same quantum code, which we call the copy-cup gate. For any desired Λ, we can construct several families of quantum codes that support gates in the Λ-th level with various asymptotic code parameters.
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