Square-lattice TQD code[1]
Description
A square-lattice Clifford-stabilizer realization of the \(2+1\)D \(l=m=n=1\) cubic theory code, equivalently the Type-III \(\mathbb{Z}_2^3\) TQD phase. The square-lattice TQD code places three qubits on every edge and provides an intermediate non-Pauli encoding for fault-tolerant non-Clifford operations on 2D topological codes [1–3].
For each of the three qubit colors, every face supports a weight-four Pauli-\(Z\) plaquette generator. Every vertex supports a Clifford generator consisting of Pauli-\(X\) operators on the four incident qubits of one color and two \(CZ\) gates acting on nearby qubits of the other two colors [3; Fig. 1]. Vertex generators of different colors need not commute on the full Hilbert space, but they commute within the simultaneous \(+1\) eigenspace of the plaquette generators [1,2].
A syndrome-extraction circuit follows by slicing the Type-III Dijkgraaf-Witten path integral on a cubic spacetime cellulation. The circuit augments three square-lattice toric-code syndrome-extraction circuits with \(CCZ\) gates that implement the cocycle twist [4,5].
Protection
For the phenomenological noise model studied on an \(L\times L\) torus, numerical simulations indicate an effective distance of approximately \(L/2\) for both just-in-time and global decoding of \(X\)-like errors, as well as exponential suppression of the logical error rate with \(L\) below threshold [3].Transversal and Permutation-Based Gates
On a square-lattice patch with three gapped boundaries encoding one logical qubit, a constant-depth automorphism circuit implements a logical \(T^\dagger\) gate [2].Decoding
A minimum-weight perfect-matching just-in-time decoder commits the required \(X\) corrections using only the syndrome history available at each time step [3].A completing-the-loop and graph-reduction heuristic uses the committed \(X\) corrections to reweight the subsequent global decoder for twisted \(Z\) errors [3].Fault Tolerance
Code switching between a folded surface code and the square-lattice TQD code, combined with just-in-time decoding and the logical \(T^\dagger\) gate, fault-tolerantly prepares a logical \(T\) magic state in \(O(d)\) rounds [2].Threshold
Under equal-rate phenomenological Pauli-\(X\) and plaquette-measurement noise, the matching-based just-in-time decoder has threshold \(2.51\pm0.31\%\), compared with \(2.90\pm0.12\%\) for a global decoder [3].For the full phenomenological noise model with equal \(X\)-like and \(Z\)-like error rates, just-in-time \(X\) decoding followed by the reweighted global \(Z\) decoder has threshold \(2.17\pm0.17\%\), compared with \(1.81\pm0.11\%\) for naive unheralded \(Z\) decoding [3].Cousins
- Hexagonal \(CZ\) code— The square-lattice TQD code and the hexagonal \(CZ\) code are distinct microscopic lattice realizations of the same Type-III \(\mathbb{Z}_2^3\) TQD phase [1,3,5,6].
- Brickwork \(XS\) stabilizer code— The square-lattice TQD code and the brickwork \(XS\) stabilizer code are distinct microscopic codes realizing the same Type-III \(\mathbb{Z}_2^3\) TQD phase [3,5].
- Dihedral \(G=D_m\) quantum-double code— The square-lattice TQD code realizes the same topological order as the \(G=D_4\) member of the dihedral quantum-double code family [1,7].
- Kitaev surface code— Code switching by gauging measurements connects a folded surface code to the square-lattice TQD code while preserving the encoded logical state [2].
Primary Hierarchy
References
- [1]
- P.-S. Hsin, R. Kobayashi, and G. Zhu, “Non-Abelian Self-Correcting Quantum Memory and Transversal Non-Clifford Gate Beyond the n \({}^{\text{1/3}}\) Distance Barrier”, PRX Quantum 6, (2025) arXiv:2405.11719 DOI
- [2]
- Anonymous, “Clifford hierarchy stabilizer codes: Transversal non-Clifford gates and magic”, Physical Review Letters (2026) arXiv:2511.02900 DOI
- [3]
- J. C. M. de la Fuente, N. Feldman, J. Eisert, and A. Bauer, “High-threshold decoding of non-Pauli codes for 2D universality”, (2026) arXiv:2604.02033
- [4]
- A. Bauer, “Low-overhead non-Clifford fault-tolerant circuits for all non-chiral abelian topological phases”, Quantum 9, 1673 (2025) arXiv:2403.12119 DOI
- [5]
- M. Davydova, A. Bauer, J. C. Magdalena de la Fuente, M. Webster, D. J. Williamson, and B. J. Brown, “Universal Fault-Tolerant Quantum Computation in 2D without Getting Tied in Knots”, Physical Review X 16, (2026) arXiv:2503.15751 DOI
- [6]
- B. Yoshida, “Topological phases with generalized global symmetries”, Physical Review B 93, (2016) arXiv:1508.03468 DOI
- [7]
- M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
Page edit log
- Victor V. Albert (2026-07-25) — most recent
Cite as:
“Square-lattice TQD code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/square_lattice_tqd, arXiv:2606.11484