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Planar Quantum Low-Density Parity-Check Codes with Open Boundaries

Zijian Liang (梁子健)1, Jens Niklas Eberhardt2,*, and Yu-An Chen (陳昱安)1,†

  • *Contact author: mail@jenseberhardt.com
  • †Contact author: yuanchen@pku.edu.cn
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PRX Quantum 6, 040330 – Published 12 November, 2025

DOI: https://doi.org/10.1103/qv65-vmzr

Abstract

Although high-threshold and low-overhead quantum low-density parity-check (qLDPC) codes, such as bivariate bicycle (BB) codes, can reduce the physical-qubit cost by an order of magnitude compared to the Kitaev toric code, their torus layout remains difficult for physical implementation. In this work, we introduce the first systematic procedure to convert BB codes into fully planar, open-boundary qLDPC codes, preserving their performance. We present planar code families with logical dimensions 6≤k≤13, e.g., ⟦78,6,6⟧, ⟦107,7,7⟧, ⟦268,8,12⟧, ⟦405,9,15⟧, ⟦348,10,13⟧, ⟦450,11,15⟧, ⟦386,12,12⟧, and ⟦362,13,11⟧, all with geometrically local weight-6 stabilizers. Allowing weight-8 stabilizers produces a ⟦282,12,14⟧ code, exhibiting an efficiency metric (kd2/n) an order of magnitude higher than the surface code. The construction combines boundary-anyon condensation with the “lattice-grafting” optimization, yielding high-performance qLDPC codes natively compatible with planar hardware architectures. It also uncovers Sierpinski-type fractal logical operators the distance of which scales with the fractal area on finite lattices. These planar qLDPC codes provide an implementable route to resource-efficient, high-threshold fault tolerance and a flexible framework for future code design on realistic two-dimensional hardware.

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Popular Summary

Quantum computers promise breakthroughs in science and technology, but their fragile nature requires powerful error correction. The surface code, today’s leading approach, is simple to implement but demands millions of physical qubits, creating a major barrier to building useful machines.

In this work, we introduce a new family of quantum error-correcting codes that dramatically reduce this overhead while preserving the same hardware-friendly, two-dimensional layout. By combining insights from topological order with a new optimization method called lattice grafting, we develop the first systematic way to design planar quantum low-density parity-check codes with open boundaries. This adapts the recently developed bivariate bicycle codes—originally defined on a torus—to open square lattices, making them more practical for experimental realization. These codes protect quantum information nearly an order of magnitude more efficiently than the surface code while remaining compatible with existing chip architectures. Notably, we also find that the logical operators of these codes form fractal patterns resembling Sierpinski triangles, offering new analytical insights into how code distance scales.

Our results not only chart a realistic path toward resource-efficient quantum computers but also uncover deep connections between quantum error correction, condensed-matter physics, and the mathematics of fractals and operator algebras.

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PRX Quantum 6, 040330– Published 12 November, 2025
  • Received 4 July 2025
  • Revised 21 August 2025
  • Accepted 7 October 2025
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