While planar toric codes provide fault tolerant quantum memory on flat two dimensional surfaces, mycelial networks naturally grow in hyperbolic three dimensional architectures with negative curvature. We extend bio topological quantum memory theory to hyperbolic surface codes, demonstrating four critical advantages. First, exponentially increased stabiliser to qubit ratios emerge from hyperbolic growth geometry. Second, code distances scale as exponential functions of radius rather than linear functions seen in planar codes. Third, error thresholds exceed two per cent compared to approximately one per cent for toric codes. Fourth, natural fault tolerance arises through geometric redundancy inherent to hyperbolic tessellations. Mycelial networks in Basidiomycota exhibit hyperbolic branching patterns that map directly to hyperbolic tessellations such as seven triangle and eight triangle tilings. Physical qubits encoded at junction points form logical qubits protected by genus g surface codes, where network topology determines code parameters. We calculate threshold improvements for code distances three, five, and seven in hyperbolic geometries and demonstrate thermodynamic syndrome extraction remains viable despite increased connectivity. This framework suggests room temperature quantum computing may achieve superior error correction by exploiting biological hyperbolic geometries rather than engineered planar architectures. Combined with recent experimental validation of mycelial bioelectronics at ambient temperatures, hyperbolic codes may provide the geometric advantage necessary to cross error correction thresholds that remain inaccessible to planar implementations.
While planar toric codes provide fault tolerant quantum memory on flat two dimensional surfaces, mycelial networks naturally grow in hyperbolic three dimensional architectures with negative curvature. We extend bio topological quantum memory theory to hyperbolic surface codes, demonstrating four critical advantages. First, exponentially increased stabiliser to qubit ratios emerge from hyperbolic growth geometry. Second, code distances scale as exponential functions of radius rather than linear functions seen in planar codes. Third, error thresholds exceed two per cent compared to approximately one per cent for toric codes. Fourth, natural fault tolerance arises through geometric redundancy inherent to hyperbolic tessellations. Mycelial networks in Basidiomycota exhibit hyperbolic branching patterns that map directly to hyperbolic tessellations such as seven triangle and eight triangle tilings. Physical qubits encoded at junction points form logical qubits protected by genus g surface codes, where network topology determines code parameters. We calculate threshold improvements for code distances three, five, and seven in hyperbolic geometries and demonstrate thermodynamic syndrome extraction remains viable despite increased connectivity. This framework suggests room temperature quantum computing may achieve superior error correction by exploiting biological hyperbolic geometries rather than engineered planar architectures. Combined with recent experimental validation of mycelial bioelectronics at ambient temperatures, hyperbolic codes may provide the geometric advantage necessary to cross error correction thresholds that remain inaccessible to planar implementations.
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(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.
Quantum error correction typically requires cryogenic temperatures and complex ancilla qubit architectures. We propose that mycelial networks, the filamentous structures of fungi, naturally implement topological quantum error correction codes at room temperature. Physical qubits correspond to bistable electrical conductance states at hyphal junctions. Network topology maps to stabilizer code structure, with plaquette and vertex operators defined on the hyphal graph. Syndrome extraction occurs through thermodynamic equilibration rather than ancilla measurements, exploiting thermal gradients to project onto code space without external measurement apparatus. We demonstrate that biological error rates of 1-5% may permit fault-tolerant operation if code thresholds can be achieved through geometric optimization. Recent experimental validation of room-temperature quantum coherence in bacterial systems and demonstrated electrical interfaces with mycelial networks provide the foundation for this framework. If successful, bio-topological quantum memory would enable quantum computing at ambient conditions with 100-fold energy reduction compared to cryogenic approaches, scalable through biological growth rather than nanofabrication.
Realizing practical quantum computers requires quantum error correction, but the most widely implemented approach, the planar surface code, demands a substantial physical qubit overhead. Here, we demonstrate that partitioning quantum processors into manageable, flat modules with non-local inter-module connections provides a natural solution. By wiring sparse, static boundary connections between Euclidean planar modules, we construct modular hyperbolic surface and color codes, based on new families of semi-hyperbolic codes. Using modular syndrome extraction circuits and efficient neural network and matching decoders, our circuit-level simulations show that this modular memory design can reduce the physical qubit overhead by tenfold or more compared to the surface code, even under elevated inter-module seam error rates. Projecting to larger system sizes, we find modular codes with encoding rate k/n=1/16 and distance d≥22, exceeding both the rate and distance of the [[288,12,18]] two-gross bivariate bicycle code while using mostly nearest-neighbor gates within planar modules, and implying an over 30× overhead reduction relative to surface codes. We develop fault-tolerant walking circuits for implementing logic via code automorphisms, which we co-design with low-weight, highly symmetrical logical bases and a modular extractor system that is ≈4.5× smaller than the code itself. Combining these methods, our modular memory can be used as dense storage in a universal fault-tolerant architecture, unlocking the efficiency of high-rate codes without sacrificing the fabrication advantages of planar modules.
The planar surface code protects quantum information robustly but pays for its two-dimensional layout with an encoding rate that vanishes as the distance grows, so fault-tolerant memories require hundreds to thousands of physical qubits per logical qubit. A recent proposal constructs modular hyperbolic surface and color codes by partitioning a quantum processor into flat Euclidean modules and wiring sparse, static boundary connections between them, yielding semi-hyperbolic codes with constant encoding rate [1,4]. In this paper we provide an independent, fully explicit quantitative analysis of that proposal. Taking the reported parameters — encoding rate k/n=1/16 and distance d≥22 — as inputs, we compute that a twelve-logical-qubit modular memory requires nmod=192 physical qubits, versus 11100 data qubits for twelve distance-22 rotated surface-code patches, a raw overhead reduction of 57.8×; the reduction remains ≥30× even if the true modular implementation uses up to 370 physical qubits. We further develop a seam-noise mixture model and compute the seam-error budget under which the modular advantage survives: with a seam fraction f=0.1443, seam links may carry up to γ≈2.5× the intra-module error rate before a tenfold advantage at matched distance is exhausted. We compare against the [[288,12,18]] bivariate bicycle code (1.5× fewer qubits for the modular code, at strictly larger distance) and situate the proposal within the low-overhead QEC landscape. All arithmetic is shown step by step; all projections are labeled with stated assumptions. ## 1. Introduction Quantum error correction (QEC) is the accepted route to fault-tolerant quantum computation, but the leading practical scheme — the planar surface code — has an encoding rate k/n that tends to zero as the code distance d grows. For a target logical error rate, the required physical-to-logical qubit ratio grows roughly quadra Full text and updates: papers.qnfo.org/papers/low-overhead-quantum-error-correction-with-boundary-connected-planar-modules-a-r/
Fault-tolerant measurement-based quantum computing (MBQC) provides a compelling framework for fault-tolerant quantum computation, in which quantum information is processed through single-qubit measurements on a three-dimensional entangled resource known as cluster state. To date, this resource has been predominantly studied on Euclidean lattices, most notably in the Raussendorf-Harrington-Goyal (RHG) construction, which underlies topological fault tolerance in MBQC. In this work, we introduce the hyperbolic cluster state, a generalization of the three-dimensional cluster state to negatively curved geometries, obtained via the foliation of periodic hyperbolic lattices. We present an explicit construction of hyperbolic cluster states and investigate their fault-tolerant properties under a realistic circuit-level depolarizing noise model. Using large-scale numerical simulations, we perform memory experiments to characterize their logical error rates and decoding performance. Our results demonstrate that hyperbolic cluster states exhibit a fault-tolerance threshold comparable to that of the Euclidean RHG cluster state, while simultaneously supporting a constant encoding rate in the thermodynamic limit. This represents a substantial improvement in qubit overhead relative to conventional cluster-state constructions. These findings establish hyperbolic geometry as a powerful and experimentally relevant resource for scalable, fault-tolerant MBQC and open new avenues for leveraging negative curvature in quantum information processing.
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. Under ancilla-based Entangling Measurement (EM3) noise, all three families achieve a threshold of ∼1.5%. With a native pair-measurement depolarizing model (SDEM3), thresholds are ∼1.0--1.2%. For heralded photon loss, the {8,3} family achieves ∼8.5--9%, exceeding the planar honeycomb threshold of ∼6.3%. In the multi-parameter SPOQC-2 noise model, the {8,3} codes achieve a 2D fault-tolerant area 2.2× that of the surface code compiled to pair measurements. We present the first photon loss and SPOQC-2 thresholds for hyperbolic Floquet codes.