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arXiv:2602.17969

Distributed Hyperbolic Floquet Codes under Depolarizing and Erasure Noise

Aygul Azatovna Galimova

quant-phSubmitted Feb 20, 2026• Announced Feb 23, 2026• 0 citations
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Abstract

Distributing qubits across quantum processing units (QPUs) connected by shared entanglement enables scaling beyond monolithic architectures. Hyperbolic Floquet codes use only weight-2 measurements and are good candidates for distributed quantum error correcting codes. 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 and distribute them across QPUs via spectral bisection. The $\{10,3\}$ and $\{12,3\}$ families are new to hyperbolic Floquet codes. We simulate these distributed codes under four noise models: depolarizing, SDEM3, correlated EM3, and erasure. With depolarizing noise ($p_{\text{local}} = 0.03\%$), fine-grained codes achieve non-local pseudo-thresholds up to 3.0\% for $\{8,3\}$, 3.0\% for $\{10,3\}$, and 1.75\% for $\{12,3\}$. Correlated EM3 yields pseudo-thresholds up to 0.75\% for $\{8,3\}$, 0.75\% for $\{10,3\}$, and 0.50\% for $\{12,3\}$; crossing-based thresholds from same-$k$ families are ${\sim}1.75$--$2.9\%$ across all tessellations. Using the SDEM3 model, fine-grained codes achieve distributed pseudo-thresholds of 1.75\% for $\{8,3\}$, 1.25\% for $\{10,3\}$, and 1.00\% for $\{12,3\}$. Under erasure noise motivated by spin-optical architectures, thresholds at 1\% local loss are 35--40\% for $\{8,3\}$, 30--35\% for $\{10,3\}$, and 25--30\% for $\{12,3\}$.

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Distributing qubits across quantum processing units (QPUs) connected by shared entanglement enables scaling beyond monolithic architectures. Hyperbolic Floquet codes use only weight-2 measurements and are good candidates for distributed quantum error correcting codes. 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 and distribute them across QPUs via spectral bisection. The {10,3}\{10,3\}{10,3} and {12,3}\{12,3\}{12,3} families are new to hyperbolic Floquet codes.
We simulate these distributed codes under four noise models: depolarizing, SDEM3, correlated EM3, and erasure. With depolarizing noise (plocal=0.03%p_{\text{local}} = 0.03\%plocal​=0.03%), fine-grained codes achieve non-local pseudo-thresholds up to 3.0% for {8,3}\{8,3\}{8,3}, 3.0% for {10,3}\{10,3\}{10,3}, and 1.75% for {12,3}\{12,3\}{12,3}. Correlated EM3 yields pseudo-thresholds up to 0.75% for {8,3}\{8,3\}{8,3}, 0.75% for {10,3}\{10,3\}{10,3}, and 0.50% for {12,3}\{12,3\}{12,3}; crossing-based thresholds from same-kkk families are ∼1.75{\sim}1.75∼1.75--2.9%2.9\%2.9% across all tessellations. Using the SDEM3 model, fine-grained codes achieve distributed pseudo-thresholds of 1.75% for {8,3}\{8,3\}{8,3}, 1.25% for {10,3}\{10,3\}{10,3}, and 1.00% for {12,3}\{12,3\}{12,3}. Under erasure noise motivated by spin-optical architectures, thresholds at 1% local loss are 35--40% for {8,3}\{8,3\}{8,3}, 30--35% for {10,3}\{10,3\}{10,3}, and 25--30% for {12,3}\{12,3\}{12,3}.
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.
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
We extend the circuit-level erasure noise model and Wang et al.\ quadratic expansion fitting of Chang et al.\ from planar surface codes to hyperbolic CSS surface codes. Under Chang et al.'s noise models, the {8,3}\{8,3\}{8,3} Bolza fine-grained family threshold reaches 3.6%3.6\%3.6% under the general-Pauli models (which coincide at temporal resolution η=1η= 1η=1) and 4.7%4.7\%4.7% under the tailored spatially perfect model at Re=1R_e = 1Re​=1 (pure erasure). The corresponding erasure-to-Pauli ratios (5.0×5.0\times5.0× and 6.5×6.5\times6.5×) match the surface code values to within 5%5\%5%. Per-observable crossing-point analysis at Re=1R_e = 1Re​=1 (Models 1--3) independently yields an erasure-to-Pauli ratio of 5.3×5.3\times5.3×. These results establish that the erasure advantage extends to hyperbolic codes.
Quantum computing offers significant speedups, but the large number of physical qubits required for quantum error correction introduces engineering challenges for a monolithic architecture. One solution is to distribute the logical quantum computation across multiple small quantum computers, with non-local operations enabled via distributed Bell states. Previous investigations of distributed quantum error correction have largely focused on the surface code, which offers good error suppression but poor encoding rates, with each surface code instance only able to encode a single logical qubit. In this work, we argue that hyperbolic Floquet codes are particularly well-suited to distributed quantum error correction for two reasons. Firstly, their hyperbolic structure enables a high number of logical qubits to be stored efficiently. Secondly, the fact that all measurements are between pairs of qubits means that each measurement only requires a single Bell state. Under the circuit-level noise model, we demonstrate through simulations that distributed hyperbolic Floquet codes offer good performance with achievable local and non-local fidelities of approximately 99.97% and 99%, respectively. This shows that distributed quantum error correction is not only possible but also efficiently realisable.
Quantum fault tolerance is a prerequisite to harness the full potential of quantum computers. However, achieving fault tolerance remains a major challenge since it requires optimizing tradeoffs between several intertwined theoretical and experimental parameters, including the encoding rate, code distance, parity-check weights, code planarity, and routing. Surface codes remain strong candidates for fault tolerance due to their simple geometric structure, established decoding methods, locality, and experimental attainability. In this work, we present finite families of hyperbolic surface codes that leverage the optimal efficiency scaling kd2/n=C(log⁡k)2kd^2/n = C (\log{k})^2kd2/n=C(logk)2 attainable by surface codes with bounded local geometry. We show that optimizing the periodic identifications can double the code distance and thereby increase the code's efficiency by a factor of four, at fixed qubit count, encoding rate, and parity-check weight. We also address the routing problem of hyperbolic surface codes by introducing a topology-aware algorithm that partitions a hyperbolic code into bounded planar modules while reducing long-range communications demand. Circuit-level Monte Carlo simulations yield finite-size threshold estimates for both modular and monolithic layouts, showing that tripling the long-range gate error probability mildly lowers, but does not eliminate, the estimated error-correction thresholds. Together, these results strengthen the case for hyperbolic surface codes as structured and highly efficient large-scale quantum error correction codes for future fault-tolerant architectures.
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.