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Tile Codes: High-Efficiency Quantum Codes on a Lattice with Boundary

Vincent Steffan1,*,†, Shin Ho Choe1,*, Nikolas P. Breuckmann2, Francisco Revson Fernandes Pereira1, and Jens Niklas Eberhardt3

  • *These authors contributed equally to this work.
  • †Contact author: vincent.steffan@meetiqm.com
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Phys. Rev. Lett. 135, 170601 – Published 21 October, 2025

DOI: https://doi.org/10.1103/l4mx-l3xx

Abstract

We introduce tile codes, a simple yet powerful way of constructing quantum codes that are local on a planar 2D lattice. Tile codes generalize the usual surface code by allowing for a bit more flexibility in terms of locality and stabilizer weight. Our construction does not compromise on the fact that the codes are local on a lattice with open boundary conditions. Despite its simplicity, we use our construction to find codes with parameters [[288,8,12]] using weight-6 stabilizers and [[288,8,14]] using weight-8 stabilizers, outperforming all previously known constructions in this direction. Allowing for a slightly higher nonlocality, we find a [[512,18,19]] code using weight-8 stabilizers, which outperforms the rotated surface code by a factor of more than 12. Our approach provides a unified framework for understanding the structure of codes that are local on a 2D planar lattice and offers a systematic way to explore the space of possible code parameters. In particular, due to its simplicity, the construction naturally accommodates various types of boundary conditions and stabilizer configurations, making it a versatile tool for quantum error-correction code design.

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  25. Note that our convention is to label both the horizontal and vertical edges touching the origin of the box in (T1) with (0,0).

  26. Consider an X tile with support (x,y) shifted by (a,b) overlapping with a Z tile at (x+a,y+b). By condition (T2), the unshifted X tile must have support [(B−1)−x−a,(B−1)−y−b]. Thus, the shifted X tile has support [(B−1)−x,(B−1)−y], which by (T2) is also in the Z tile’s support. This symmetry ensures that all overlaps are even.

  27. On the infinite lattice Z×Z, tiles with condition (T2) give rise to a Pauli Hamiltonian with commuting terms. It turns out [20, 28] that the groundspace of this Hamiltonian exhibits topological order only if (T2) holds. Practically, this means that (T2) is necessary if we want these tile codes to have distance scaling with the dimensions of the layout.

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Outline

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Phys. Rev. Lett. 135, 170601– Published 21 October, 2025
  • Received 6 May 2025
  • Revised 4 August 2025
  • Accepted 20 August 2025
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