Algorithmic Construction of Gapped Boundaries for Topological Stabilizer Codes: A Route to Low-Overhead Planar qLDPC Codes
Abstract
Quantum low-density parity-check (qLDPC) codes promise low-overhead fault-tolerant quantum memory, but many leading constructions are naturally defined on periodic lattices, whereas scalable quantum hardware favors planar architectures with open boundaries. In this work, we develop a general operator-algebraic framework and computational algorithms for constructing all gapped boundaries and defects of two-dimensional topological Pauli stabilizer codes. Starting from a truncated bulk code, our method identifies boundary gauge operators, boundary anyons, and their string operators, and then constructs microscopic boundary Hamiltonians through anyon condensation and topological-order completion, thereby eliminating local logical operators near the boundary. We establish a one-to-one bulk-boundary correspondence between bulk anyons and excitations of the anomalous boundary subsystem code, enabling fusion rules, topological spins, and braiding statistics to be extracted directly from the boundary operator algebra. For translation-invariant codes, Laurent-polynomial representations reduce these tasks to finite matrix computations based on Hermite and Smith normal forms, enabling automated constructions for Z_{d} qudits of both prime and composite dimension. We demonstrate the framework on toric, color, double-semion, six-semion, and anomalous three-fermion codes, and apply it to bivariate bicycle codes, revealing large logical dimensions and unusually long translation periods for anyon strings. By systematically converting bulk topological data into explicit boundary and defect Hamiltonians, our framework can serve as a foundation for constructing low-overhead planar qLDPC codes with open boundaries and for engineering fault-tolerant logical operations.
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BibTeX
@article{梁子健2026algorithmic,
title = {{Algorithmic Construction of Gapped Boundaries for Topological Stabilizer Codes: A Route to Low-Overhead Planar qLDPC Codes}},
author = {Zijian Liang (梁子健) and Bowen Yang (杨博闻) and Joseph T. Iosue and Yu-An Chen (陳昱安)},
journal = {PRX Quantum},
year = {2026},
month = oct,
doi = {10.1103/989h-xjqv},
url = {http://link.aps.org/doi/10.1103/989h-xjqv},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Algorithmic Construction of Gapped Boundaries for Topological Stabilizer Codes: A Route to Low-Overhead Planar qLDPC Codes
AU - Zijian Liang (梁子健)
AU - Bowen Yang (杨博闻)
AU - Joseph T. Iosue
AU - Yu-An Chen (陳昱安)
PY - 2026
DA - 2026/10/23
JO - PRX Quantum
DO - 10.1103/989h-xjqv
UR - http://link.aps.org/doi/10.1103/989h-xjqv
AB - Quantum low-density parity-check (qLDPC) codes promise low-overhead fault-tolerant quantum memory, but many leading constructions are naturally defined on periodic lattices, whereas scalable quantum hardware favors planar architectures with open boundaries. In this work, we develop a general operator-algebraic framework and computational algorithms for constructing all gapped boundaries and defects of two-dimensional topological Pauli stabilizer codes. Starting from a truncated bulk code, our method identifies boundary gauge operators, boundary anyons, and their string operators, and then constructs microscopic boundary Hamiltonians through anyon condensation and topological-order completion, thereby eliminating local logical operators near the boundary. We establish a one-to-one bulk-boundary correspondence between bulk anyons and excitations of the anomalous boundary subsystem code, enabling fusion rules, topological spins, and braiding statistics to be extracted directly from the boundary operator algebra. For translation-invariant codes, Laurent-polynomial representations reduce these tasks to finite matrix computations based on Hermite and Smith normal forms, enabling automated constructions for Z_{d} qudits of both prime and composite dimension. We demonstrate the framework on toric, color, double-semion, six-semion, and anomalous three-fermion codes, and apply it to bivariate bicycle codes, revealing large logical dimensions and unusually long translation periods for anyon strings. By systematically converting bulk topological data into explicit boundary and defect Hamiltonians, our framework can serve as a foundation for constructing low-overhead planar qLDPC codes with open boundaries and for engineering fault-tolerant logical operations.
SN - 2691-3399
ER -