Anyon Theory and Topological Frustration of High-Efficiency Quantum Low-Density Parity-Check Codes

Quantum low-density parity-check (QLDPC) codes offer a promising path to low-overhead fault-tolerant quantum computation but lack systematic strategies for exploration. In this Letter, we establish a topological framework for studying the bivariate-bicycle codes, a prominent class of QLDPC codes tailored for real-world quantum hardware. Our framework enables the investigation of these codes through universal properties of topological orders. In addition to efficient characterizations using Gr\"obner bases, we also introduce a novel algebraic-geometric approach based on the Bernstein--Khovanskii--Kushnirenko theorem. This approach allows us to analytically determine how the topological order varies with the generic choices of bivariate-bicycle codes under toric layouts. Novel phenomena are unveiled, including topological frustration, where ground-state degeneracy on a torus deviates from the total anyon number, and quasi-fractonic mobility, where anyon movement violates energy conservation. We demonstrate their intrinsic link to symmetry-enriched topological orders and derive an efficient method for generating finite-size codes. Furthermore, we extend the connection between anyons and logical operators using Koszul complex theory. Our Letter provides a rigorous theoretical basis for exploring the fault tolerance of QLDPC codes and deepens the interplay among topological order, quantum error correction, and advanced algebraic structures.