Recent advances in quantum hardware relax the strict geometric-locality constraints traditionally imposed on quantum error-correcting codes, motivating interest in high-rate quantum low-density parity-check (qLDPC) codes. At the same time, color codes provide a particularly rich setting for fault-tolerant quantum computation, underlying protocols such as single-shot error correction and self-correcting quantum computation. Hyperbolic color codes provide a class of high-rate qLDPC codes that also retain the structural features of color codes relevant to fault-tolerant quantum computation. However, previous constructions of hyperbolic color codes have achieved at most logarithmic code distance. In this work, we construct hyperbolic color codes with both constant rate and polynomial distance by building on arithmetic hyperbolic manifolds that support polynomial-distance hyperbolic toric codes. Our construction applies in arbitrary dimension \(D\geq 4\). In even dimensions, the resulting type-\(D/2\) color codes have constant encoding rate and polynomial distance, while in other cases the number of logical qubits and the code distance both exhibit polynomial scaling. We further derive explicit exponents for polynomial lower bounds as functions of the dimension and code type. These results establish a family of hyperbolic color codes simultaneously achieving constant rate and polynomial distance and providing a testbed to explore fault-tolerant quantum computation protocols that combine high-rate quantum codes with the structural advantages of color codes.
Recent advances in quantum hardware relax the strict geometric-locality constraints traditionally imposed on quantum error-correcting codes, motivating interest in high-rate quantum low-density parity-check (qLDPC) codes. At the same time, color codes provide a particularly rich setting for fault-tolerant quantum computation, underlying protocols such as single-shot error correction and self-correcting quantum computation. Hyperbolic color codes provide a class of high-rate qLDPC codes that also retain the structural features of color codes relevant to fault-tolerant quantum computation. However, previous constructions of hyperbolic color codes have achieved at most logarithmic code distance. In this work, we construct hyperbolic color codes with both constant rate and polynomial distance by building on arithmetic hyperbolic manifolds that support polynomial-distance hyperbolic toric codes. Our construction applies in arbitrary dimension (D\geq 4). In even dimensions, the resulting type-(D/2) color codes have constant encoding rate and polynomial distance, while in other cases the number of logical qubits and the code distance both exhibit polynomial scaling. We further derive explicit exponents for polynomial lower bounds as functions of the dimension and code type. These results establish a family of hyperbolic color codes simultaneously achieving constant rate and polynomial distance and providing a testbed to explore fault-tolerant quantum computation protocols that combine high-rate quantum codes with the structural advantages of color codes.
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.
We study parallel fault-tolerant quantum computing for families of homological quantum low-density parity-check (LDPC) codes defined on 3-manifolds with constant or almost-constant encoding rate. We derive a generic formula for a transversal T gate on color codes defined on general 3-manifolds, which acts as collective non-Clifford logical gates on any triplet of logical qubits with their logical- X membranes having a Z 2 triple intersection at a single point. The triple-intersection number is a topological invariant, which also arises in the path integral of the emergent higher symmetry operator in a topological quantum field theory (TQFT): the Z 2 3 gauge theory. Moreover, the transversal S gate of the color code corresponds to a higher-form symmetry in TQFT supported on a codimension-1 submanifold, giving rise to exponentially many addressable and parallelizable logical gates. A construction of constant-depth circuits of the above logical gates via cup-product cohomology operation is also presented for three copies of identical toric codes on arbitrary 3-manifolds. We have developed a generic formalism to compute the triple-intersection invariants for 3-manifolds, with the structure encoded into an interaction hypergraph which determines the logical gate property and also corresponds to the hypergraph magic state that can be injected into the code without distillation (“”). We also study the scaling of the Betti number and systoles with volume for various 3-manifolds, which translates to the encoding rate and distance. We further develop three types of LDPC codes supporting such logical gates: (1) A quasi-hyperbolic code from the product of 2D hyperbolic surface and a circle, with almost-constant rate k / n = O ( 1 / log ( n ) ) and O ( log ( n ) ) distance; (2) A homological fiber-bundle code from twisting the product by an isometry of the surface based on the construction by Freedman-Meyer-Luo, with O ( 1 / log 1 2 ( n ) ) rate and O ( log 1 2 ( n ) ) distance; (3) A specific family of 3D hyperbolic codes: the Torelli mapping-torus code, constructed from mapping tori of a pseudo-Anosov element in the Torelli subgroup, which has constant rate while the distance scaling is currently unknown. We then show a generic constant-overhead scheme for applying a parallelizable universal gate set with the aid of logical- X measurements.
Quantum low-density parity-check (LDPC) codes are an important class of quantum error correcting codes. In such codes, each qubit only affects a constant number of syndrome bits, and each syndrome bit only relies on some constant number of qubits. Constructing quantum LDPC codes is challenging. It is an open problem to understand if there exist good quantum LDPC codes, i.e. with constant rate and relative distance. Furthermore, techniques to perform fault-tolerant gates are poorly understood. We present a unified way to address these problems. Our main results are a) a bound on the distance, b) a bound on the code dimension and c) limitations on certain fault-tolerant gates that can be applied to quantum LDPC codes. All three of these bounds are cast as a function of the graph separator of the connectivity graph representation of the quantum code. We find that unless the connectivity graph contains an expander, the code is severely limited. This implies a necessary, but not sufficient, condition to construct good codes. This is the first bound that studies the limitations of quantum LDPC codes that does not rely on locality. As an application, we present novel bounds on quantum LDPC codes associated with local graphs in D -dimensional hyperbolic space.
A central goal in quantum error correction is to reduce the overhead of fault-tolerant quantum computing by increasing noise thresholds and reducing the number of physical qubits required to sustain a logical qubit. We introduce a potential path towards this goal based on a family of dynamically generated quantum error correcting codes that we call "hyperbolic Floquet codes.'' These codes are defined by a specific sequence of non-commuting two-body measurements arranged periodically in time that stabilize a topological code on a hyperbolic manifold with negative curvature. We focus on a family of lattices for n qubits that, according to our prescription that defines the code, provably achieve a finite encoding rate ( 1 / 8 + 2 / n ) while still requiring only two-body measurements. Similar to hyperbolic surface codes, the distance of the code at each time-step scales at most logarithmically in n . The family of lattices we choose indicates that this scaling is achievable in practice. We develop and benchmark an efficient matching-based decoder that provides evidence of a threshold near 0.1% in a phenomenological noise model and 0.25% in an entangling measurements noise model. Utilizing weight-two check operators and a qubit connectivity of 3, one of our hyperbolic Floquet codes uses 400 physical qubits to encode 52 logical qubits with a code distance of 8, i.e., it is a [ [ 400 , 52 , 8 ] ] code. At small error rates, comparable logical error suppression to this code requires 5x as many physical qubits (1924) when using the honeycomb Floquet code with the same noise model and decoder.
Quantum computation has shown advantages in several problems over the corresponding classical algorithms. The noisy intermediate-size\n quantum devices with dozens of qubits make it more promising in the\n past decade, as the quantum advantage was demonstrated\n experimentally. Before entering the era of scalable quantum\n computation, one has to resolve the errors in a quantum many-body\n system, which is inevitable due to the environment interaction during\n quantum control processes. The goal of quantum error correction is to\n reduce such errors and increase decoherence time. The most successful candidates of quantum error correction codes are topological codes, especially surface codes, which were discovered by\n Alexei Kitaev. The ordered phase of the 2D\n Ising model on a torus ensures that toric codes are fault-tolerant\n below a critical error probability. Other than the FT threshold,\n the topological codes feature efficient encoding and\n decoding, local measurements, but suffer from asymptotically zero\n code rate. To encode more logical qubits with finite resources, one can loosen the condition on locality and extend to a broader class of quantum\n low-density parity-check (LDPC)codes. There are many known algebraic\n constructions for such codes, but only a precious few of them have\n finite code rates and meet the fault-tolerant condition:\n the stabilizer weight is bounded and the distance scales at least\n logarithmically with the code size. In this thesis, I construct the\n higher-dimensional quantum hypergraph product (HQHP) codes, which\n generalize quantum hypergraph product (QHP) codes and toric codes in\n all dimensions. The HQHP codes projected into a single space gives\n subsystem product codes, which can then be gauge fixed to\n concatenated codes or homological product codes. Those include some\n common CSS codes, like Shor’s codes, Bacon Shor’s codes, and\n subsystem QHP codes. The HQHP codes can be mapped to the tensor\n product of chain complexes, which provides an algebraic framework to\n construct quantum LDPC codes with finite code rates, square root\n distances, FT thresholds and single-shot properties with redundant\n checks. Meanwhile, their rich connection to other codes are very\n instructive and may lead to further optimizations. Regarding the\n remained procedures towards fault-tolerant universal quantum\n computation through quantum LDPC codes, I will discuss the\n fault-tolerant condition for each code, including distance,\n stabilizer weight, decoding, fault-tolerant gates and measurement\n protocols.\n
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/