Hyperbolic and Semi-Hyperbolic Floquet Codes for Photonic Quantum Computing
Aygul Azatovna Galimova
TL;DR
Hyperbolic and semi-hyperbolic Floquet codes are constructed from $\{8,3\}$, $\{10,3\}$, and $\{12,3\}$ tessellations via the Wythoff kaleidoscopic construction with the Low-Index Normal Subgroups (LINS) algorithm and evaluated under four noise models.
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.