Floquet Codes from Derived Semi-Regular Hyperbolic Tessellations on Orientable and Non-Orientable Surfaces

Abstract: In this paper, we construct several new quantum Floquet codes on compact, orientable, as well as non-orientable surfaces. In order to obtain such codes, we identify these surfaces with hyperbolic polygons and examine hyperbolic semi-regular tessellations on such surfaces. The method of construction presented here generalizes similar constructions concerning hyperbolic Floquet codes on connected and compact surfaces with genus $g \geq 2$. A performance analysis and an investigation of the asymptotic behavior of these codes are also presented.

• The paper introduces a novel method for constructing quantum Floquet codes using derived semi-regular hyperbolic tessellations on both orientable and non-orientable compact surfaces.
• It details clipping and incenter derivation techniques to explicitly calculate code parameters such as length, dimension, and distance across various tessellation types and genera.
• The study reveals trade-offs between encoded qubit counts and error-correction performance, offering enhanced rates and robustness compared to traditional regular tessellation-based codes.

Floquet Codes from Derived Semi-Regular Hyperbolic Tessellations on Orientable and Non-Orientable Surfaces

Introduction and Motivation

The paper addresses the explicit construction and analysis of quantum Floquet codes based on derived semi-regular hyperbolic tessellations on both orientable and non-orientable compact surfaces of genus $g\geq 2$ (2603.29811). Floquet codes—dynamic subsystem stabilizer codes with time-dependent logical operators and low-weight measurement checks—are known to increase resilience against noise in quantum systems. Existing approaches focused on tessellations that are regular and typically restricted to orientable manifolds. This work introduces generative techniques for semi-regular hyperbolic tessellations, producing an expanded landscape of Floquet code families, including those on non-orientable topologies, and provides a rigorous parameter analysis, including length, dimension, and distance.

Tessellation Structures and Mathematical Framework

The framework builds on the geometry of Riemann surfaces, the combinatorics of tessellations, and algebraic coding theory. The authors define tessellations $T=(V, E, F)$ where $V$, $E$, $F$ are vertices, edges, and faces, emphasizing locally finite uniform tilings with regular polygons. Regular tessellations are described by Schläfli symbols $\{p, q\}$, representing tilings with $p$-gons meeting $q$ at each vertex. In hyperbolic geometry, such tilings exist iff $1/p + 1/q < 1/2$. Semi-regular tessellations, denoted $[n_1, n_2, \dots, n_m]$, generalize this by allowing multiple regular polygons cyclically arranged at vertices.

The core geometric quantities are derived using the Gauss-Bonnet theorem and explicit combinatorial formulae for the number of faces, edges, and vertices on closed surfaces of given genus. These results are used to construct tessellation-based code lattices over the corresponding surfaces. For non-orientable surfaces, new existence and generation results for semi-regular patterns are developed, enabling code design beyond the orientable case.

Construction of Floquet Codes

Floquet codes are constructed by assigning qubits to tessellation elements, with check operators prescribed by tessellation geometry. The construction leverages trivalent (3-colorable) tilings, obtained both directly and via dualization, and applies two specific tessellation derivation techniques:

• Clipping Derivation: Starting from a regular $T=(V, E, F)$0 tessellation, regions near vertices are clipped to create $T=(V, E, F)$1 semi-regular tessellations.
• Incenter Derivation: New faces are constructed by drawing segments from centers of original faces and their adjacent polygons, yielding $T=(V, E, F)$2 tilings.

For quantum code construction, the hyperbolic tessellations are projected onto compact surfaces of genus $T=(V, E, F)$3, leading to code parameters determined by tessellation combinatorics and the surface Euler characteristic.

Code Parameter Analysis

The code parameters—length $T=(V, E, F)$4, dimension $T=(V, E, F)$5, and distance $T=(V, E, F)$6—are calculated explicitly for each tessellation and genus. For orientable surfaces, $T=(V, E, F)$7; for non-orientable, $T=(V, E, F)$8. The code length equals the number of vertices in the tessellation, which is determined by closed-form expressions as functions of $T=(V, E, F)$9, $V$0, and $V$1. The distance computation is nontrivial: it relies on the hyperbolic geodesic lengths of homologically nontrivial cycles, which serve as logical operators, and is lower-bounded using the prescribed metric properties of the tiling.

The paper systematically tabulates code parameters for numerous tessellation types and genera, explicitly displaying the achievable $V$2 codes over both orientable and non-orientable topologies. The analysis includes the coding rate $V$3, the normalized distance $V$4, and the rate $V$5 (motivated by local decodability considerations), providing a comparative basis for code class performance.

Orientable vs. Non-orientable Surface Codes

A central result is the extension of code constructions to non-orientable surfaces, achieved by leveraging fundamental polygon equivalences and Euler characteristic arguments. A theorem establishes conditions where codes on orientable surfaces can be matched with equivalent codes on non-orientable surfaces (of even genus), with identical parameters, check weights, and geometric embeddings. However, non-orientable surfaces of odd genus lead to genuinely novel codes with, in many instances, superior $V$6 rates.

Comparative tabulation demonstrates that non-orientable codes achieve higher $V$7 but (for fixed $V$8) lower $V$9 than their orientable counterparts, indicating a classical trade-off between the encoded qubit count and error-correcting capability as measured by normalized distance. In the limit of large genus, both $E$0 asymptotically converge, but the difference in $E$1 remains nontrivial for practical parameter regimes.

Asymptotics and Performance

An in-depth asymptotic analysis shows that by tuning the tessellation parameters $E$2 and genus $E$3, the families of semi-regular hyperbolic Floquet codes can interpolate between “Euclidean-like” (quasi-regular) and “hyperbolic” code classes, achieving rates up to $E$4 as $E$5 for select families (e.g., $E$6). The codes derived from $E$7 and $E$8 yield quasi-Euclidean rates approaching the honeycomb code ($E$9). The inclusion of both large- and small-edged polygons within the same tessellation enhances the flexibility and density of the available code families.

Explicit parameter tables evidence strong $F$0 for certain families, including those surpassing the honeycomb and toric code rates for the corresponding surfaces, highlighting the effectiveness of the derived semi-regular approach in practical code constructions.

Implications and Future Directions

The work substantially enriches the space of Floquet codes by systematically generalizing hyperbolic tessellation techniques, introducing semi-regular and non-orientable classes, and analyzing trade-offs and rates of these codes in detail. Practically, these findings provide new tools for quantum error correction architectures where hardware geometry or noise constraints favor non-trivial topology or low-weight measurements. Theoretically, the construction techniques and asymptotic behaviors provide a richer understanding of the relationship between geometry, topology, and quantum code performance.

Future research directions include decoding algorithm optimization for these new code classes, hardware embedding studies leveraging non-orientable topologies, exploration of residual symmetries in semi-regular tilings for code switchability and code concatenation, and the pursuit of threshold analysis under physical noise models tailored to dynamic Floquet architectures.

Conclusion

This paper introduces a broad framework for constructing and analyzing Floquet codes on both orientable and non-orientable surfaces using derived semi-regular hyperbolic tessellations. By providing explicit techniques for tessellation derivation, detailed parameter calculation, and asymptotic analysis, it establishes a foundation for exploiting surface topology and generalized tessellation structures in the design of resilient, efficient quantum codes. The results demonstrate both improved code class flexibility and enhanced parameter regimes compared to prior work restricted to regular or orientable tessellations.