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arXiv:2608.07431

Designer Codes from GALA: Compact, Self-Dual, and Rate-1/2 QEC on Reconfigurable Atom Arrays

Willers Yang, Casey Duckering, Arpit Dua

quant-phSubmitted Aug 7, 2026• Announced Aug 10, 2026• 0 citations
v1 · Aug 7, 2026v2 · Oct 5, 2026
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Abstract

High rate quantum low-density parity-check codes on reconfigurable neutral-atom arrays can reduce the overhead of quantum error correction, but near-term devices support only hundreds of qubits with limited reconfigurability from a few crossed acousto-optic deflectors (AOD). A practical code must be compact in addition to low-overhead, with checks and logical gates mapping onto hardware-compatible physical instructions. We introduce the GALA codes, or Group-Action Lifts with Active orthogonality, that 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, reaching $1/2$ rate with above-weight distance, while the large abelian factor $C_m$ supplies symmetries that give code automorphisms and explicit, simple AOD move schedules. Hardware compatibility and logical capability thereby become customizable inputs to a code search rather than properties verified post-hoc, making GALA designer codes by construction. The GALA family contains several previously discovered 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. Our search yields a compact self-dual $[[132, 30, 12]]$ with a small number of $4$-cycles (almost girth-6) and below $10^{-8}$ logical error rate (LER) for memory at $10^{-3}$ physical error rate, 3.1ms syndrome-extraction cycle and transversal Clifford gates; a girth-6, rate-$1/2$ $[[672, 336, 12]]$ with a 6.76ms cycle with below $10^{-10}$ LER (extrapolated) and rate-$1/2$ barrier-breaking $[[1752, 880, 14]]$ and $[[2232, 1120, 16]]$ with exactly certified distances greater than check weights and all smaller than previously known hardware compatible rate-1/2 codes.

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Quantum error correction at ultra-low overhead

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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 $1/2$ while maintaining a pseudo-threshold exceeding $0.4\%$ 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 $X$- and $Z$-type checks in parallel with $12$ 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 $[[2844,1426,18]]$ encodes $1{,}426$ distance-$18$ logical qubits, achieving an extrapolated logical error rate of $2.6\times10^{-16}$ ($1.9\times10^{-31}$) per logical qubit per cycle, assuming the physical error rate of $0.1\%$ ($0.01\%$). By comparison, a bivariate bicycle code implementation would require more than $68{,}000$ 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.

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Low-space-overhead fault-tolerant quantum computation requires not only high-rate quantum error-correcting codes but also space-efficient implementations of logical operations. Transversal and fold-transversal gates are promising since they limit error propagation and require no additional qubits. However, the logical operations they enable are typically restricted, and a central challenge is to construct codes that combine a complete set of logical Clifford gates with favorable code parameters. In this work, we introduce a family of quantum codes with an asymptotically constant encoding rate and sublogarithmically growing stabilizer weight, while supporting the entire logical Clifford group using only transversal and fold-transversal gates. Our construction is based on classical codes whose code spaces are absolutely irreducible Steinberg modules of their Tanner-graph automorphism groups. Taking hypergraph products of these classical codes yields quantum codes for which all logical Clifford operations can be synthesized from a fixed set of transversal and fold-transversal gates. Moreover, the slow growth of the stabilizer weight enables a high error-correcting performance in small instances. These results provide a path toward fault-tolerant quantum computation with low space overhead using transversal and fold-transversal gates.

quant-pharXiv:2609.37699•Sep 30, 2026•0 citations
High rate quantum low-density parity-check codes on reconfigurable neutral-atom arrays can reduce the overhead of quantum error correction, but near-term devices support only hundreds of qubits with limited reconfigurability from a few crossed acousto-optic deflectors (AOD). A practical code must be compact in addition to low-overhead, with checks and logical gates mapping onto hardware-compatible physical instructions. We introduce the GALA codes, or Group-Action Lifts with Active orthogonality, that 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, reaching 1/21/21/2 rate with above-weight distance, while the large abelian factor CmC_mCm​ supplies symmetries that give code automorphisms and explicit, simple AOD move schedules. Hardware compatibility and logical capability thereby become customizable inputs to a code search rather than properties verified post-hoc, making GALA designer codes by construction. The GALA family contains several previously discovered 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. Our search yields a compact self-dual [[132,30,12]][[132, 30, 12]][[132,30,12]] with a small number of 444-cycles (almost girth-6) and below 10−810^{-8}10−8 logical error rate (LER) for memory at 10−310^{-3}10−3 physical error rate, 3.1ms syndrome-extraction cycle and transversal Clifford gates; a girth-6, rate-1/21/21/2 [[672,336,12]][[672, 336, 12]][[672,336,12]] with a 6.76ms cycle with below 10−1010^{-10}10−10 LER (extrapolated) and rate-1/21/21/2 barrier-breaking [[1752,880,14]][[1752, 880, 14]][[1752,880,14]] and [[2232,1120,16]][[2232, 1120, 16]][[2232,1120,16]] with exactly certified distances greater than check weights and all smaller than previously known hardware compatible rate-1/2 codes.
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.
Despite significant progress on quantum low-density parity-check (qLDPC) codes, building qLDPC processors that are high-rate, high-throughput, hardware-friendly, and fast-to-decode remains a challenge. We introduce mitten codes, a family of qLDPC processor codes of encoding rate 20%20\%20% and check weight 999, based on non-abelian groups. Their non-abelian structure evades distance bounds constraining abelian counterparts, allowing mitten codes to reach distance 181818 and beyond with just a few hundred data qubits. The logical operators of a mitten code are related by the group action, yielding a modular, low-overhead logical toolkit: full Clifford operations follow from bridging two reusable seed surgery gadgets or from a single fixed extractor. Furthermore, qLDPC processors based on mitten codes support high-rate surgery that executes many logical measurements in parallel, and parallel magic-state injection into all logical qubits at once. Under circuit-level noise, with our fast decoder, the [ ⁣[300,60,14] ⁣][\![300,60,14]\!][[300,60,14]] mitten code achieves, without extrapolation, a block logical error rate of ∼10−11{\sim}10^{-11}∼10−11 per round at 0.1%0.1\%0.1% physical error rate (PER), while the [ ⁣[975,195,≤24] ⁣][\![ 975,195,\leq 24 ]\!][[975,195,≤24]] code reaches ∼10−8{\sim}10^{-8}∼10−8 at 0.4%0.4\%0.4% PER. Decoding 151515 billion surgery experiments on the [ ⁣[540,108,18] ⁣][\![540,108,18]\!][[540,108,18]] code at 0.1%0.1\%0.1% PER, we observe only two logical failures, demonstrating a qLDPC processor capable of running ∼1010{\sim}10^{10}∼1010 logical operations. Our decoder is compatible with sub-millisecond average latency per logical cycle, sufficient for real-time decoding on neutral atom hardware. Discovered by an end-to-end design pipeline built on sQetch, a distance estimator orders of magnitude faster than existing tools, and mapping efficiently onto near-term neutral atom and superconducting hardware, mitten codes open a practical path toward fault-tolerant quantum computation.
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.
We study the structure and transversal logical capabilities of stabilizer quantum error correcting codes. Among our results, we identify universal lower bounds on circuit depth to generate a full logical Clifford algebra, and develop novel constructions of logical transversal gates including a new depth-one transversal phase S‾\mathrm{\overline{S}}S gate in the rotated surface code and a depth-one intra-block CZ‾\mathrm{\overline{CZ}}CZ gate in the 2D-toric code that generalizes to all odd distances and all lengths L≥3L\ge3L≥3, respectively. Finally, we construct a high-rate non-LDPC CSS code family with parameters [[n,n,Θ(nβ)]][[n,\sqrt{n},Θ({n^β})]][[n,n​,Θ(nβ)]] where β≈0.2823β\approx 0.2823β≈0.2823 in one demonstrated case, that provably possesses a constant-depth complete 2-local transversal logical Clifford basis instruction set architecture (ISA) composed of all individually targeted S‾\mathrm{\overline{S}}S, SHS‾=X\mathrm{\overline{SHS}} = \sqrt{X}SHS=X​, and CZ‾\mathrm{\overline{CZ}}CZ gates. This ISA is depth-one for certain subfamilies that we design and generally constant-depth under certain conditions. The code family is built from a small code with parameters [[n0,2,d0]][[n_0, 2, d_0]][[n0​,2,d0​]], and is tunable in the standard way: it tiles out to form utility-scale logical qubit counts, and it scales up through concatenation to achieve higher distances and error suppression. We show that this construction preserves the depth-one complete transversal logical Clifford basis ISA when composed with these commuting construction actions, inheriting structure from the core codes so that at scale the complete logical Clifford basis ISA remains depth-one up to depth-two addressable operations between tiled cores. We call these Quantum Logic Codes.
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.
Low-space-overhead fault-tolerant quantum computation requires not only high-rate quantum error-correcting codes but also space-efficient implementations of logical operations. Transversal and fold-transversal gates are promising since they limit error propagation and require no additional qubits. However, the logical operations they enable are typically restricted, and a central challenge is to construct codes that combine a complete set of logical Clifford gates with favorable code parameters. In this work, we introduce a family of quantum codes with an asymptotically constant encoding rate and sublogarithmically growing stabilizer weight, while supporting the entire logical Clifford group using only transversal and fold-transversal gates. Our construction is based on classical codes whose code spaces are absolutely irreducible Steinberg modules of their Tanner-graph automorphism groups. Taking hypergraph products of these classical codes yields quantum codes for which all logical Clifford operations can be synthesized from a fixed set of transversal and fold-transversal gates. Moreover, the slow growth of the stabilizer weight enables a high error-correcting performance in small instances. These results provide a path toward fault-tolerant quantum computation with low space overhead using transversal and fold-transversal gates.