Hyperbolic and Semi-Hyperbolic Floquet Codes for Photonic Quantum Computing

Abstract: Tailoring error correcting codes to the structure of the physical noise can reduce the overhead of fault-tolerant quantum computation. Hyperbolic Floquet codes use only weight-2 measurements and can be implemented directly on hardware with native pair measurements. We construct hyperbolic and semi-hyperbolic Floquet codes from ${8,3}$, ${10,3}$, and ${12,3}$ tessellations via the Wythoff kaleidoscopic construction with the Low-Index Normal Subgroups (LINS) algorithm. The ${10,3}$ and ${12,3}$ families are new to hyperbolic Floquet codes. We evaluate these codes under four noise models: phenomenological, ancilla Entangling Measurement (EM3), Single-step Depolarizing EM3 (SDEM3), and erasure. Under phenomenological noise, specific-logical threshold crossings occur near $p_e \approx 0.3$--$0.5\%$ for ${8,3}$ ($k=6$--$56$) and $0.15$--$0.2\%$ for ${10,3}$ ($k=12$--$146$). EM3 ancilla noise yields a threshold of ${\sim}1.5\%$ for all three families. SDEM3 is a depolarizing noise model motivated by Majorana tetron architectures; fine-grained codes achieve thresholds of ${\sim}1.0$--$1.2\%$ for all three families. The erasure model captures detected photon loss on spin-optical links; fine-grained codes achieve erasure thresholds of ${\sim}8.5$--$9\%$ for ${8,3}$, ${\sim}7$--$8\%$ for ${10,3}$, and ${\sim}6.5$--$8\%$ for ${12,3}$. Photon loss is the dominant error source in photon-mediated quantum computing. Under the full three-parameter SPOQC-2 noise model, the ${8,3}$ codes achieve a 2D fault-tolerant area $2.2\times$ that of the surface code compiled to pair measurements while encoding $k = 10$ logical qubits. In a companion paper, we evaluate the same code families in a distributed setting.