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  • Open Access

Planar Fault-Tolerant Circuits for Non-Clifford Gates on the 2D Color Code

Andreas Bauer1,* and Julio C. Magdalena de la Fuente2,†

  • *Contact author: andib@mit.edu
  • †Contact author: jm@juliomagdalena.de
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PRX Quantum 7, 020367 – Published 23 June, 2026

DOI: https://doi.org/10.1103/p7c9-x1m9

Abstract

We introduce a family of scalable planar fault-tolerant circuits that implement logical non-Clifford operations on a 2D color code, such as a logical T gate or a logical non-Pauli measurement that prepares a magic |T⟩ state. The circuits are relatively simple, consisting only of physical T gates, CX gates, and few-qubit measurements. They can be implemented with an array of qubits on a 2D chip with nearest-neighbor couplings and no wire crossings. The construction is based on a spacetime path integral representation of a non-Abelian 2+1D topological phase, which is related to the 3D color code. We turn the path integral into a circuit by expressing it as a spacetime ZX tensor network and then traversing it in some chosen time direction. We describe in detail how fault tolerance is achieved using a “just-in-time” decoding strategy, for which we repurpose and extend state-of-the-art color-code matching decoders.

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Popular Summary

Fault-tolerant quantum computing with 2D topological codes is attractive because it matches planar hardware, but universal computation still hinges on implementing at least one non-Clifford ingredient (notably the T gate or T-magic-state primitives) in a way that does not dominate resources. We present a family of fully planar, geometrically local fault-tolerant protocols (“twisted color circuits”) that realize such non-Clifford logical operations while staying within a 2D nearest-neighbor layout and using only standard components—single-qubit T gates, CX gates, and few-qubit Pauli measurements—so they are directly compatible with chip-scale architectures. Our construction starts from a spacetime path-integral description related to the 3D color code, translates it into a ZX tensor-network form, and then compiles it into explicit circuits with well-defined boundaries and interfaces back to the 2D color code. Because T gates make “decode-at-the-end” insufficient, we incorporate a just-in-time decoding approach that continually processes syndromes during execution, leveraging matching-based decoders and casting the core subroutines as efficient hypergraph-matching problems (with reductions to coupled cellular matching where possible). We also provide a supporting mathematical framework (color cohomology) that clarifies defect/syndrome constraints, boundary types, and the relation of the underlying spacetime model to a known non-Abelian topological gauge theory, which in turn guides both circuit design and decoding.

Article Text

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PRX Quantum 7, 020367– Published 23 June, 2026
  • Received 30 September 2025
  • Revised 23 February 2026
  • Accepted 24 April 2026
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