Quantum error correction codes defined on hyperbolic lattices leverage the unique geometric properties of the hyperbolic space to enhance the performance of quantum error correction. By embedding qubits in hyperbolic lattices, these codes achieve higher encoding rates and lower qubit overhead compared to those defined on conventional Euclidean lattices. Building on recent advances in hyperbolic crystallography, we introduce a unified framework for the systematic construction and scalable benchmarking of CSS quantum error correction codes on hyperbolic lattices. A central component of this framework is the Hyperbolic Cycle Basis algorithm, which employs graph-theoretic methods to efficiently identify all plaquette cycles (parity-check supports) and nontrivial cycles (logical operators). This enables scalable and automated benchmarking of a broad class of CSS codes defined on hyperbolic geometries. We apply this framework to construct and simulate two representative hyperbolic quantum error correction codes (HQECCs), evaluating key performance metrics such as encoding rate, error threshold, and code distance for different sublattices. While HQECCs serve as concrete examples, the framework can be adapted to a wide range of CSS codes, including those with more intricate stabilizer structures such as Floquet codes. This work establishes a foundation for systematic exploration and benchmarking of CSS codes on hyperbolic lattices, paving the way toward practical, high-performance quantum error correction.
Quantum error correction codes defined on hyperbolic lattices leverage the unique geometric properties of the hyperbolic space to enhance the performance of quantum error correction. By embedding qubits in hyperbolic lattices, these codes achieve higher encoding rates and lower qubit overhead compared to those defined on conventional Euclidean lattices. Building on recent advances in hyperbolic crystallography, we introduce a unified framework for the systematic construction and scalable benchmarking of CSS quantum error correction codes on hyperbolic lattices. A central component of this framework is the Hyperbolic Cycle Basis algorithm, which employs graph-theoretic methods to efficiently identify all plaquette cycles (parity-check supports) and nontrivial cycles (logical operators). This enables scalable and automated benchmarking of a broad class of CSS codes defined on hyperbolic geometries. We apply this framework to construct and simulate two representative hyperbolic quantum error correction codes (HQECCs), evaluating key performance metrics such as encoding rate, error threshold, and code distance for different sublattices. While HQECCs serve as concrete examples, the framework can be adapted to a wide range of CSS codes, including those with more intricate stabilizer structures such as Floquet codes. This work establishes a foundation for systematic exploration and benchmarking of CSS codes on hyperbolic lattices, paving the way toward practical, high-performance quantum error correction.
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≥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.
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 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.
Reducing the qubit overhead of quantum error correction is a central challenge for scalable fault-tolerant quantum computing. Recent ultra-high-rate quantum codes offer a promising route toward this goal, with some constructions requiring as few as two physical data qubits per logical qubit. However, systematic principles for navigating the tradeoffs among encoding rate, distance, check weight, and blocklength remain lacking. Here, we develop and analyze principles for exploring this design space, and use them to design compact code constructions with improved performance. We develop code templates based on a pair-partition construction and a halving transformation that further reduces blocklength. Motivated by ensemble analysis of the degree distributions, we identify column weight as a key design parameter: increasing the column weight enables larger distances at compact blocklengths, at the cost of heavier checks. We find that at physical error rates of 0.1%, the benefits of increased distance often outweigh the penalty associated with heavier checks. Applying this framework, we identify numerous compact codes with favorable parameters, including [[90,21,11]], [[140,31,15]], and [[200,43,20]] non-CSS codes with check weight 10. Moreover, we develop symmetry-informed strategies for identifying low-weight logical bases. These results provide systematic strategies for designing ultra-high-rate quantum codes and navigating their Pareto frontier.
We introduce a new class of qubit codes that we call Evenbly codes, building on a previous proposal of hyperinvariant tensor networks. Its tensor network description consists of local, non-perfect tensors describing CSS codes interspersed with Hadamard gates, placed on a hyperbolic {p,q} geometry with even q≥4 , yielding an infinitely large class of subsystem codes. We construct an example for a {5,4} manifold and describe strategies of logical gauge fixing that lead to different rates k/n and distances d , which we calculate analytically, finding distances which range from d=2 to d∼n2/3 . Investigating threshold performance under erasure, depolarizing, and pure Pauli noise channels, we find that the code exhibits a depolarizing noise threshold of about 19.1% in the code-capacity model and 50% for pure Pauli and erasure channels under suitable gauges. We also test a constant-rate version with k/n=0.125 , finding excellent error resilience (about 40%) under the erasure channel. Recovery rates for these and other settings are studied both under an optimal decoder as well as a more efficient but non-optimal greedy decoder. We also consider generalizations beyond the CSS tensor construction, compute error rates and thresholds for other hyperbolic geometries, and discuss the relationship to holographic bulk/boundary dualities. Our work indicates that Evenbly codes may show promise for practical quantum computing applications.
Quantum Error Correction (QEC) is essential for overcoming the fragility of qubits, enabling reliable and scalable quantum computing by mitigating the effects of noise, decoherence, and operational errors. This chapter explores QEC's foundational principles, including error types, noise impact, and syndrome measurement techniques for identifying and correcting errors. It covers key codes such as Shor and Steane, which provide basic error protection, and advanced codes like concatenated codes, surface codes, and quantum Low-Density Parity-Check (LDPC) codes. Concatenated codes employ hierarchical encoding to exponentially reduce logical error rates, while surface codes, including Toric codes, utilize topological properties of qubits arranged on a 2D lattice to detect and correct errors with high fault tolerance and scalability. Quantum LDPC codes leverage sparse parity-check matrices to offer efficient and scalable error correction, closely approaching theoretical performance limits. The chapter concludes by exploring current research efforts to optimize QEC codes.