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arXiv:2602.10423

Erasure Thresholds for Hyperbolic and Semi-Hyperbolic Surface Codes

Aygul Azatovna Galimova

quant-phSubmitted Feb 11, 2026• Announced Feb 12, 2026• 0 citations• Revised Mar 18, 2026 (v2)
v1 · Feb 11, 2026v2 · Mar 18, 2026
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Abstract

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 $\{8,3\}$ Bolza fine-grained family threshold reaches $3.6\%$ under the general-Pauli models (which coincide at temporal resolution $η= 1$) and $4.7\%$ under the tailored spatially perfect model at $R_e = 1$ (pure erasure). The corresponding erasure-to-Pauli ratios ($5.0\times$ and $6.5\times$) match the surface code values to within $5\%$. Per-observable crossing-point analysis at $R_e = 1$ (Models 1--3) independently yields an erasure-to-Pauli ratio of $5.3\times$. These results establish that the erasure advantage extends to hyperbolic codes.

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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.
Studying the threshold behavior of surface codes under biased noise models is an active area of research. Previous work Tuckett et al. (2018), using an optimal tensor-network decoder, demonstrated that replacing ZZZ-type stabilizers with YYY-type stabilizers significantly improves the surface code threshold under code-capacity level dephasing noise. In this work, we construct and study a ZYZYZY surface code by replacing the XXX-type stabilizers with YYY-type stabilizers. We compare it with the standard ZXZXZX surface code under circuit-level Pauli-XXX biased noise, with and without an additional gate-based XXXXXX crosstalk noise. We find that for the ZXZXZX surface code, the XXX-memory threshold increases monotonically with bias while the ZZZ-memory threshold decreases and saturates. For the ZYZYZY surface code, the YYY-memory threshold is nearly constant across all bias values. The ZZZ-memory thresholds of the ZXZXZX and ZYZYZY codes are consistent within the uncertainty. Adding XXXXXX crosstalk reduces the ZZZ-memory threshold beyond the fitting uncertainty while leaving the XXX memory threshold largely unaffected. The choice of CNOT ordering redistributes threshold performance between the two logical memories. Our work extends prior observations from code-capacity level noise to circuit-level noise. It also indicates the need for decoders capable of jointly reasoning over correlated syndrome information so that tailored stabilizer structures could be fully utilized for quantum error correction.
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
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}.
We show how a hyperbolic surface code could be used for overhead-efficient quantum storage. We give numerical evidence for a noise threshold of $1.3 \\% $ for the 4,5\\{4,5\\}4,5-hyperbolic surface code in a phenomenological noise model (as compared with $2.9 \\% $ for the toric code). In this code family, parity checks are of weight 4 and 5, while each qubit participates in four different parity checks. We introduce a family of semi-hyperbolic codes that interpolate between the toric code and the 4,5\\{4,5\\}4,5-hyperbolic surface code in terms of encoding rate and threshold. We show how these hyperbolic codes outperform the toric code in terms of qubit overhead for a target logical error probability. We show how Dehn twists and lattice code surgery can be used to read and write individual qubits to this quantum storage medium.
Abstract We show how a hyperbolic surface code could be used for overhead-efficient quantum storage. We give numerical evidence for a noise threshold of 1.3 % for the { 4 , 5 } -hyperbolic surface code in a phenomenological noise model (as compared with 2.9 % for the toric code). In this code family, parity checks are of weight 4 and 5, while each qubit participates in four different parity checks. We introduce a family of semi-hyperbolic codes that interpolate between the toric code and the { 4 , 5 } -hyperbolic surface code in terms of encoding rate and threshold. We show how these hyperbolic codes outperform the toric code in terms of qubit overhead for a target logical error probability. We show how Dehn twists and lattice code surgery can be used to read and write individual qubits to this quantum storage medium.
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.