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Universal Fault-Tolerant Quantum Computation in 2D without Getting Tied in Knots

Margarita Davydova1,2,3, Andreas Bauer4, Julio C. Magdalena de la Fuente5, Mark Webster6,7, Dominic J. Williamson7,8, and Benjamin J. Brown2,9

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Phys. Rev. X 16, 031001 – Published 7 July, 2026

DOI: https://doi.org/10.1103/vrty-qs5h

Abstract

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.

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Popular Summary

Universal fault-tolerant quantum computation requires a practical scheme for executing logic gates with quantum error-correcting codes. We show how to perform scalable non-Clifford gates in two dimensions by temporarily transferring information from surface codes into a non-Abelian topological phase and back again. By utilizing domain walls and the spacetime path integral framework, we derive microscopic circuits that produce the action of complex logical operations such as CCZ and T gates. These protocols offer a resource-efficient alternative to magic state distillation. We also establish a threshold theorem for fault tolerance under circuit noise using a just-in-time decoder. Our findings provide an avenue for universal quantum computing by bridging the gap spacetime topological protocols and two-dimensional hardware architectures.

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Phys. Rev. X 16, 031001– Published 7 July, 2026
  • Received 15 May 2025
  • Revised 9 December 2025
  • Accepted 26 March 2026
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