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arXiv:2604.16209

Towards Ultra-High-Rate Quantum Error Correction with Reconfigurable Atom Arrays

Chen Zhao, Casey Duckering, Andi Gu, Nishad Maskara, Hengyun Zhou

quant-phcs.ITSubmitted Apr 17, 2026• Announced Apr 20, 2026• 0 citations• Revised Jun 26, 2026 (v2)
v1 · Apr 17, 2026v2 · Jun 26, 2026
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Abstract

Quantum error correction is widely believed to be essential for large-scale quantum computation, but the required qubit overhead remains a central challenge. Quantum low-density parity-check codes can substantially reduce this overhead through high-rate encodings, yet finite-size instances with practical logical error rates often achieve encoding rates only around or below $1/10$. Here, building on a recent ultra-high-rate construction by Kasai, we identify new structural conditions on the underlying affine permutation matrices that make encoding rates exceeding $1/2$ compatible with efficient implementation on reconfigurable neutral atom arrays. These conditions define a co-designed family of ultra-high-rate quantum codes that supports efficient syndrome extraction and atom rearrangement under realistic parallel control constraints. Using a hierarchical decoder with high accuracy and good throughput, we study the performance under a circuit-level noise model with $p=0.1\%$, achieving per-logical-per-round error rates of $1.3_{-0.9}^{+3.0} \times 10^{-13}$ with a $[[2304,1156,\leq 14]]$ code and $2.9_{-1.5}^{+3.1} \times 10^{-11}$ with a $[[1152,580,\leq 12]]$ code. We compare these codes against a heuristic Pareto frontier for finite-blocklength codes relating block length, encoding rate, and logical error rates, and find that our codes lie near the frontier. These results approach the teraquop regime, highlighting the promise of this code family for practical ultra-high-rate quantum error correction.

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Achieving useful fault-tolerant applications requires co-designing the quantum error-correcting code for footprint, logical instruction set, and hardware compatibility simultaneously. Recent breakthroughs in ultra-high-rate codes have the potential to reduce footprint of large scale quantum applications but their practical utility is limited without an addressable logical instruction set and efficient implementation on near term hardware. Reconfigurable neutral atom arrays, for example, support only hundreds of qubits with constrained reconfigurability from a few crossed acousto-optic deflectors (AOD). We introduce the GALA codes, or Group-Action Lifts with Active orthogonality, that allows such co-design by construction. GALA lifts over a product group $G = H_k \times C_m$ (or $H_k \ltimes C_m^k$); the small non-abelian factor $H_k$ supplies active orthogonality that improves qubit overhead, reaching 1/2 encoding rate with above-weight distance, while the large abelian factor $C_m$ supplies symmetries that give code automorphisms and explicit, simple AOD move schedules. Logical addressibility and hardware compatibility follow directly from construction, rather than verified post-hoc. The GALA family contains several known rate-1/2 Kasai codes of Ref.[arXiv:2601.08824, arXiv:2604.16209] while exposing simpler parameter bounds, logical operations, and ZX-dual variants with AOD-compatible fold-transversal Clifford gates. In particular, we find a compact self-dual $[[132,30,12]]$ with $1.0^{+0.9}_{-0.6}\times 10^{-9}$ per-round per-qubit logical error rate at $10^{-3}$ physical error rate, equipped with 3.1 ms syndrome-extraction cycles and an addressable logical instruction set; a ZX-dual, girth-6, rate-1/2 $[[672,336,12]]$ with a 6.76 ms cycles; and barrier-breaking $[[1752,880,14]]$ and $[[2232,1120,16]]$ smaller than previously known hardware-compatible rate-1/2 codes.

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Quantum low-density parity-check (qLDPC) codes can achieve high encoding rates and good code distance scaling, providing a promising route to low-overhead fault-tolerant quantum computing. However, the long-range connectivity required to implement such codes makes their physical realization challenging. Here, we propose a hardware-efficient scheme to perform fault-tolerant quantum computation with high-rate qLDPC codes on reconfigurable atom arrays, directly compatible with recently demonstrated experimental capabilities. Our approach utilizes the product structure inherent in many qLDPC codes to implement the non-local syndrome extraction circuit via atom rearrangement, resulting in effectively constant overhead in practically relevant regimes. We prove the fault tolerance of these protocols, perform circuit-level simulations of memory and logical operations with these codes, and find that our qLDPC-based architecture starts to outperform the surface code with as few as several hundred physical qubits at a realistic physical error rate of $10^{-3}$. We further find that less than 3000 physical qubits are sufficient to obtain over an order of magnitude qubit savings compared to the surface code, and quantum algorithms involving thousands of logical qubits can be performed using less than $10^5$ physical qubits. Our work paves the way for explorations of low-overhead quantum computing with qLDPC codes at a practical scale, based on current experimental technologies.

arXiv:2308.08648•Aug 16, 2023•6 citations
Quantum error correction is widely believed to be essential for large-scale quantum computation, but the required qubit overhead remains a central challenge. Quantum low-density parity-check codes can substantially reduce this overhead through high-rate encodings, yet finite-size instances with practical logical error rates often achieve encoding rates only around or below 1/101/101/10. Here, building on a recent ultra-high-rate construction by Kasai, we identify new structural conditions on the underlying affine permutation matrices that make encoding rates exceeding 1/21/21/2 compatible with efficient implementation on reconfigurable neutral atom arrays. These conditions define a co-designed family of ultra-high-rate quantum codes that supports efficient syndrome extraction and atom rearrangement under realistic parallel control constraints. Using a hierarchical decoder with high accuracy and good throughput, we study the performance under a circuit-level noise model with p=0.1%p=0.1\%p=0.1%, achieving per-logical-per-round error rates of 1.3−0.9+3.0×10−131.3_{-0.9}^{+3.0} \times 10^{-13}1.3−0.9+3.0​×10−13 with a [[2304,1156,≤14]][[2304,1156,\leq 14]][[2304,1156,≤14]] code and 2.9−1.5+3.1×10−112.9_{-1.5}^{+3.1} \times 10^{-11}2.9−1.5+3.1​×10−11 with a [[1152,580,≤12]][[1152,580,\leq 12]][[1152,580,≤12]] code. We compare these codes against a heuristic Pareto frontier for finite-blocklength codes relating block length, encoding rate, and logical error rates, and find that our codes lie near the frontier. These results approach the teraquop regime, highlighting the promise of this code family for practical ultra-high-rate quantum error correction.
Suppressing errors is the central challenge for useful large-scale quantum computing. While quantum error correction promises a viable solution to this challenge, existing codes typically suffer from trade-offs among encoding efficiency, error threshold, and hardware feasibility. Here, we introduce Cornucopia codes, a family of practical, hardware-efficient quantum low-density parity-check codes that achieve an ultra-high encoding rate exceeding 1/21/21/2 while maintaining a pseudo-threshold exceeding 0.4%0.4\%0.4% under the standard circuit-level noise model. Inspired by recent affine-permutation-based code constructions and the long-range connectivity available in reconfigurable neutral-atom arrays, we adopt a structured code geometry in which the code layout, atom rearrangement, and syndrome-extraction schedule are co-designed. This structure enables nonlocal syndrome measurements through simple, parallel atom rearrangements. A complete syndrome extraction cycle measures all XXX- and ZZZ-type checks in parallel with 121212 entangling layers, independent of the code size. The resulting threshold is comparable to those of the surface code and bivariate bicycle codes. In particular, a single code block [[2844,1426,18]][[2844,1426,18]][[2844,1426,18]] encodes 1,4261{,}4261,426 distance-181818 logical qubits, achieving an extrapolated logical error rate of 2.6×10−162.6\times10^{-16}2.6×10−16 (1.9×10−311.9\times10^{-31}1.9×10−31) per logical qubit per cycle, assuming the physical error rate of 0.1%0.1\%0.1% (0.01%0.01\%0.01%). By comparison, a bivariate bicycle code implementation would require more than 68,00068{,}00068,000 physical qubits to encode the same number of logical qubits at a comparable logical error rate. These results bring demonstrations of ultra-low-overhead quantum error correction within the reach of near-term quantum processors.
Constant-rate quantum low-density parity-check (LDPC) codes promise fault-tolerant quantum computation with constant spatial overhead in the asymptotic limit. Nonetheless, discovering finite-length code instances with good practical performance remains challenging. We introduce a new family of quantum LDPC codes with design rate 1/51/51/5 and check weight 999 that approaches the teraquop memory regime per qubit-round with several hundred physical qubits, under idling-free circuit-level noise of strength 0.1%0.1\%0.1% and GPU-accelerated Relay-belief-propagation (Relay-BP) decoding with average latencies around 1-2 ms, a regime relevant to trapped-ion and neutral-atom processors. The construction involves the balanced product of classical LDPC codes with design rate 1/21/21/2 that share non-abelian Zℓ⋊Zm\mathbb{Z}_\ell \rtimes \mathbb{Z}_mZℓ​⋊Zm​ group symmetries, which may be of independent interest for classical error correction. We build syndrome extraction circuits tailored to reconfigurable atom arrays using a simple greedy scheduler, with single-round rearrangement times around 30-60 ms using present hardware specifications, and substantial room for future improvements. We also construct logical Pauli bases that are equivariant with respect to the group symmetry, which can significantly compress the design space for code surgery. Together, these results further advance the practicality of constant-rate quantum LDPC codes for near-term, fault-tolerant quantum computers.
Achieving useful fault-tolerant applications requires co-designing the quantum error-correcting code for footprint, logical instruction set, and hardware compatibility simultaneously. Recent breakthroughs in ultra-high-rate codes have the potential to reduce footprint of large scale quantum applications but their practical utility is limited without an addressable logical instruction set and efficient implementation on near term hardware. Reconfigurable neutral atom arrays, for example, support only hundreds of qubits with constrained reconfigurability from a few crossed acousto-optic deflectors (AOD). We introduce the GALA codes, or Group-Action Lifts with Active orthogonality, that allows such co-design by construction. GALA lifts over a product group G=Hk×CmG = H_k \times C_mG=Hk​×Cm​ (or Hk⋉CmkH_k \ltimes C_m^kHk​⋉Cmk​); the small non-abelian factor HkH_kHk​ supplies active orthogonality that improves qubit overhead, reaching 1/2 encoding rate with above-weight distance, while the large abelian factor CmC_mCm​ supplies symmetries that give code automorphisms and explicit, simple AOD move schedules. Logical addressibility and hardware compatibility follow directly from construction, rather than verified post-hoc. The GALA family contains several known rate-1/2 Kasai codes of Ref.[arXiv:2601.08824, arXiv:2604.16209] while exposing simpler parameter bounds, logical operations, and ZX-dual variants with AOD-compatible fold-transversal Clifford gates. In particular, we find a compact self-dual [[132,30,12]][[132,30,12]][[132,30,12]] with 1.0−0.6+0.9×10−91.0^{+0.9}_{-0.6}\times 10^{-9}1.0−0.6+0.9​×10−9 per-round per-qubit logical error rate at 10−310^{-3}10−3 physical error rate, equipped with 3.1 ms syndrome-extraction cycles and an addressable logical instruction set; a ZX-dual, girth-6, rate-1/2 [[672,336,12]][[672,336,12]][[672,336,12]] with a 6.76 ms cycles; and barrier-breaking [[1752,880,14]][[1752,880,14]][[1752,880,14]] and [[2232,1120,16]][[2232,1120,16]][[2232,1120,16]] smaller than previously known hardware-compatible rate-1/2 codes.
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.
Quantum low-density parity-check (qLDPC) codes offer a promising route to reducing the qubit overhead of fault tolerance, with recent ultra-high-rate constructions achieving encoding rates above one half. In this work, we develop fast and parallel logical computation methods for ultra-high-rate codes. We first utilize the group symmetries of the codes to partition the logical qubits into orbits, and then apply parallel surgery methods to implement group-invariant collections of logical Pauli-product measurements. These measurements are highly programmable, as we may arbitrarily couple logical orbits as long as each measurement is repeated across the whole orbit(s). To reduce the space and time overheads, we develop a technique called chain-map convolution, which uses the group structure to boost both the spatial and temporal distances of our protocol in a highly efficient manner. As case studies, we construct distance-preserving parallel measurement gadgets for the [[256, 68, 12]], [[576, 148, 18]] pair-partition codes, and a [[1152, 580, 12]] rate-1/21/21/2 code, with only around 2.5×2.5\times2.5× space overhead and 3-4 rounds of syndrome extraction each. The same methods also enable parallel injection of cultivated magic states into high-rate codes. Finally, we extend programmability through subgroup restrictions, revealing a tradeoff between logical control and the spacetime resources required for fault tolerance and highlighting the crucial role structured parallelism plays in reducing overhead. These results provide new methods for designing qLDPC logical operations with low spacetime overhead, opening a route toward more efficient fault-tolerant quantum architectures.
Quantum low-density parity-check (qLDPC) codes can achieve high encoding rates and good code distance scaling, providing a promising route to low-overhead fault-tolerant quantum computing. However, the long-range connectivity required to implement such codes makes their physical realization challenging. Here, we propose a hardware-efficient scheme to perform fault-tolerant quantum computation with high-rate qLDPC codes on reconfigurable atom arrays, directly compatible with recently demonstrated experimental capabilities. Our approach utilizes the product structure inherent in many qLDPC codes to implement the non-local syndrome extraction circuit via atom rearrangement, resulting in effectively constant overhead in practically relevant regimes. We prove the fault tolerance of these protocols, perform circuit-level simulations of memory and logical operations with these codes, and find that our qLDPC-based architecture starts to outperform the surface code with as few as several hundred physical qubits at a realistic physical error rate of 10−310^{-3}10−3. We further find that less than 3000 physical qubits are sufficient to obtain over an order of magnitude qubit savings compared to the surface code, and quantum algorithms involving thousands of logical qubits can be performed using less than 10510^5105 physical qubits. Our work paves the way for explorations of low-overhead quantum computing with qLDPC codes at a practical scale, based on current experimental technologies.