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arXiv:2603.29811

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

Douglas F. Copatti, Giuliano G. La Guardia, Waldir S. Soares, Edson D. Carvalho, Eduardo B. Silva

quant-phcs.ITSubmitted Mar 31, 2026• Announced Apr 1, 2026• 0 citations
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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.

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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≥2g \geq 2g≥2. A performance analysis and an investigation of the asymptotic behavior of these codes are also presented.
In this paper, we construct new families of asymmetric quantum surface codes (AQSCs) over non-orientable surfaces of genus g ≥ 2 by applying tools of hyperbolic geometry. More precisely, we prove that all asymmetric quantum surface codes constructed over orientable surfaces with genus h can be reproduced, with exactly the same parameters, over a non-orientable surface with genus g = 2h. Additionally, if S is a non-orientable surface of genus g, we show that the new surface code constructed on a {p, q} tessellation of S has ratio k/n greater than the ratio of an AQSC constructed on the same {p, q} tessellation over an orientable surface of the same genus g.
Current work presents a new approach to quantum color codes on compact surfaces with genus g\geq2 using the identification of these surfaces with hyperbolic polygons and hyperbolic tessellations. We show that this method may give rise to color codes with a very good parameters and we present tables with several examples of these codes whose parameters had not been shown before. We also present a family of codes with minimum distance d=4 and the encoding rate asymptotically going to 1 while n\rightarrow\infty.
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}\{8,3\}{8,3}, {10,3}\{10,3\}{10,3}, and {12,3}\{12,3\}{12,3} tessellations via the Wythoff kaleidoscopic construction with the Low-Index Normal Subgroups (LINS) algorithm. The {10,3}\{10,3\}{10,3} and {12,3}\{12,3\}{12,3} families are new to hyperbolic Floquet codes. We evaluate these codes under four noise models. Under ancilla-based Entangling Measurement (EM3) noise, all three families achieve a threshold of ∼1.5%{\sim}1.5\%∼1.5%. With a native pair-measurement depolarizing model (SDEM3), thresholds are ∼1.0{\sim}1.0∼1.0--1.2%1.2\%1.2%. For heralded photon loss, the {8,3}\{8,3\}{8,3} family achieves ∼8.5{\sim}8.5∼8.5--9%9\%9%, exceeding the planar honeycomb threshold of ∼6.3%{\sim}6.3\%∼6.3%. In the multi-parameter SPOQC-2 noise model, the {8,3}\{8,3\}{8,3} codes achieve a 2D fault-tolerant area 2.2×2.2\times2.2× that of the surface code compiled to pair measurements. We present the first photon loss and SPOQC-2 thresholds for hyperbolic Floquet codes.
Current work presents a new approach to quantum color codes on compact surfaces with genus g≥2g \geq 2g≥2 using the identification of these surfaces with hyperbolic polygons and hyperbolic tessellations. We show that this method may give rise to color codes with a very good parameters and we present tables with several examples of these codes whose parameters had not been shown before. We also present a family of codes with minimum distance d=4d=4d=4 and the encoding rate asymptotically going to 1 while n→∞n \rightarrow \inftyn→∞.
We construct families of Floquet codes derived from color-code tilings of closed hyperbolic surfaces. These codes have weight-two check operators, a finite encoding rate and can be decoded efficiently with minimum-weight perfect matching. We also construct semi-hyperbolic Floquet codes, which have improved distance scaling, and are obtained via a fine-graining procedure. Using a circuit-based noise model that assumes direct two-qubit measurements, we show that semi-hyperbolic Floquet codes can be 48 times more efficient than planar honeycomb codes and therefore over 100 times more efficient than alternative compilations of the surface code to two-qubit measurements, even at physical error rates of 0.3% to 1%. We further demonstrate that semi-hyperbolic Floquet codes can have a teraquop footprint of only 32 physical qubits per logical qubit at a noise strength of 0.1%. For standard circuit-level depolarizing noise at p=0.1% , we find a 30 times improvement over planar honeycomb codes and a 5.6 times improvement over surface codes. Finally, we analyze small instances that are amenable to near-term experiments, including a Floquet code derived from the Bolza surface that encodes four logical qubits into 16 physical qubits. Published by the American Physical Society 2024