Single-shot magic state factories for quantum LDPC codes
Varun Menon, Rohan Mehta, Andi Gu, Andrei C. Diaconu, Daniel Bochen Tan, Xiao Xiao, Michael J. Gullans, Qian Xu, Hengyun Zhou, Mikhail D. Lukin, J. Pablo Bonilla Ataides
quant-phSubmitted Oct 7, 2026• Announced Oct 9, 2026• 0 citations
Quantum low-density parity-check (qLDPC) codes can substantially reduce the qubit overhead of fault-tolerant quantum computation, but any useful architecture based on such codes also requires an efficient mechanism to implement non-Clifford gates in a fault-tolerant manner. We introduce a space-time-efficient magic state factory for high-rate qLDPC codes by combining the transversal non-Clifford gates of recently developed tricycle codes with single-shot four-dimensional balanced product quadcycle codes. The key ingredient is an efficient "escape" mechanism that teleports postselected logical magic states from tricycle codes to larger distance quadcycle codes. Under a standard circuit-level noise model at physical error rate $10^{-3}$, our factories produce $\mathrm{CCZ}$ states at logical error rates as low as $10^{-9}$ with 10-40 times lower space-time cost than state-of-the-art magic state generation methods. The resulting states are directly encoded in high-rate blocks suitable for computation. We identify the quadcycle codes by an AI-assisted search and show that they can serve as excellent memories capable of single-shot error correction. Finally, we present a concrete implementation of the entire factory with highly efficient atom transport on reconfigurable neutral-atom quantum computers. Together, these results provide an end-to-end, practical protocol for producing and utilizing high-fidelity magic states within a high-rate qLDPC architecture, with the lowest space-time cost to date.
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17 main text pages, 6 main figures, Supplementary Information appended. Atom movement animation included in ancillary files
Quantum low-density parity-check (qLDPC) codes can substantially reduce the qubit overhead of fault-tolerant quantum computation, but any useful architecture based on such codes also requires an efficient mechanism to implement non-Clifford gates in a fault-tolerant manner. We introduce a space-time-efficient magic state factory for high-rate qLDPC codes by combining the transversal non-Clifford gates of recently developed tricycle codes with single-shot four-dimensional balanced product quadcycle codes. The key ingredient is an efficient "escape" mechanism that teleports postselected logical magic states from tricycle codes to larger distance quadcycle codes. Under a standard circuit-level noise model at physical error rate 10−3, our factories produce CCZ states at logical error rates as low as 10−9 with 10-40 times lower space-time cost than state-of-the-art magic state generation methods. The resulting states are directly encoded in high-rate blocks suitable for computation. We identify the quadcycle codes by an AI-assisted search and show that they can serve as excellent memories capable of single-shot error correction. Finally, we present a concrete implementation of the entire factory with highly efficient atom transport on reconfigurable neutral-atom quantum computers. Together, these results provide an end-to-end, practical protocol for producing and utilizing high-fidelity magic states within a high-rate qLDPC architecture, with the lowest space-time cost to date.
The preparation of high-fidelity non-Clifford (magic) states is an essential subroutine for universal quantum computation but imposes substantial space-time overhead. Magic-state factories based on high-rate and -distance quantum low-density parity-check (LDPC) codes equipped with transversal non-Clifford gates can potentially reduce these overheads significantly, by circumventing the need for multiple rounds of distillation and by producing a large number of magic states in a single code block. As a step toward realizing efficient, fault-tolerant magic-state production, we introduce a class of finite block-length quantum LDPC codes which we name tricycle codes, generalizing the well-known bicycle codes to three homological dimensions. These codes can support constant-depth physical circuits that implement logical C C Z gates between three code blocks. To construct these constant-depth C C Z circuits, we develop analytical and numerical techniques that apply to a broad class of three-dimensional homological and balanced-product codes. We further show that tricycle codes enable single-shot state preparation and error correction, leading to a highly efficient magic-state generation protocol. Numerical simulations of specific codes confirm robust performance under circuit-level noise, demonstrating a high circuit-noise threshold of > 0.5 % . With modest postselection, certain tricycle codes of block lengths of only approximately 50–100 qubits are shown to produce | C C Z ¯ ⟩ magic states with logical error rates ranging from 2 × 10 − 8 to below 3 × 10 − 11 . Finally, we construct optimal depth syndrome extraction circuits for tricycle codes and present a protocol for implementing them efficiently on a reconfigurable neutral-atom platform.
Quantum error correction offers a path from fragile physical qubits to reliable quantum computation, but the resource cost of conventional architectures remains a central obstacle to useful machines. This thesis develops a full-stack approach based on high-rate quantum low-density parity-check (qLDPC) codes, which encode many logical qubits per block and can substantially reduce fault-tolerance overhead. Its central premise is that favorable code parameters are useful only when memory, hardware, logical operations, decoding, and interconnects are designed together. Quantum memories based on hypergraph-product, lifted-product, spatially coupled, and balanced-product codes provide the underlying algebraic structure, while reconfigurable neutral-atom arrays realize their nonlocal connectivity through parallel optical control, high-fidelity Rydberg gates, and loss-resolving readout. Universal logical computation is enabled through teleportation, homomorphic gates, and finite-block-length tricycle constructions for low-cost non-Clifford resource states. Structure-aware neural decoders learn spatial, temporal, and gate-induced correlations while remaining compatible with real-time feed-forward, and high-rate entanglement-distillation protocols extend the same constant-overhead principles to quantum interconnects. Together, these results outline an integrated route toward practical fault-tolerant quantum computers built from high-rate quantum codes.
While quantum low-density parity check (qLDPC) codes are a low-overhead means of quantum information storage, it is valuable for quantum codes to possess fault-tolerant features beyond this resource efficiency. In this work, we introduce trivariate tricycle (TT) codes, qLDPC codes that combine several desirable features: high thresholds under a circuit-level noise model, partial single-shot decodability for low-time-overhead decoding, a large set of transversal Clifford gates and automorphisms within and between code blocks, and (for several sub-constructions) constant-depth implementations of a (non-Clifford) CCZ gate. TT codes are CSS codes based on a length-3 chain complex, and are defined from three trivariate polynomials, with the 3D toric code (3DTC) belonging to this construction. We numerically search for TT codes and find several candidates with improved parameters relative to the 3DTC, using up to 48× fewer data qubits as equivalent 3DTC encodings. We construct syndrome-extraction circuits for these codes and numerically demonstrate single-shot decoding in Z memory, in both phenomenological and circuit-level noise models. We also demonstrate that the codes have good performance under single-shot decoding even in X memory. Under circuit-level noise, TT codes have a crossing point of 0.3% in X memory and 1% in Z memory (with single-shot decoding). All TT codes possess several transversal CZ gates that can partially address logical qubits between two code blocks. The codes possess a large set of automorphisms that can perform Clifford gates within a code block. Finally, we establish several TT code polynomial constructions that allow for a constant-depth implementation of logical CCZ gates. We find codes with CCZ gates that use up to 5.7× fewer data qubits as equivalent-distance 3DTC encodings, and others with encoding rates as high as 1/4.
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% and check weight 9, based on non-abelian groups. Their non-abelian structure evades distance bounds constraining abelian counterparts, allowing mitten codes to reach distance 18 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]] mitten code achieves, without extrapolation, a block logical error rate of ∼10−11 per round at 0.1% physical error rate (PER), while the [[975,195,≤24]] code reaches ∼10−8 at 0.4% PER. Decoding 15 billion surgery experiments on the [[540,108,18]] code at 0.1% PER, we observe only two logical failures, demonstrating a qLDPC processor capable of running ∼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.
High-rate quantum LDPC (qLDPC) codes reduce memory overhead by densely packing many logical qubits into a single block of physical qubits. Here we extend this concept to high-rate computation by constructing \emph{batched} fault-tolerant operations that apply the same logical gate across many code blocks in parallel. By leveraging shared physical resources to execute many logical operations in parallel, these operations realize high rates in space-time and significantly reduce computational costs. For \emph{arbitrary} CSS qLDPC codes, we build batched gadgets with \emph{constant space-time overhead} (assuming fast classical computation) for (i) single-shot error correction, state preparation, and code surgeries (ii) code switching, and (iii) addressable Clifford gates. Using these batched gadgets we also construct parallel non-Clifford gates with low space-time cost. We outline principles for designing parallel quantum algorithms optimized for a batched architecture, and show in particular how lattice Hamiltonian dynamical simulations can be compiled efficiently. We also propose a near-term implementation using new self-dual Bivariate-Bicycle codes with high encoding rates (∼1/10), transversal Clifford gates, and global T gates via parallel magic state cultivation, enabling Hamiltonian simulations with a lower space-time cost than analogous surface-code protocols and low-rate qLDPC protocols. These results open new paths toward scalable quantum computation via co-design of parallel quantum algorithms and high-rate fault-tolerant protocols.
Quantum low-density parity-check (qLDPC) codes offer a promising route to scalable fault-tolerant quantum computing due to their substantially reduced footprint. However, these gains can be diluted at utility scale if we cannot also realize space-time efficient logical operations for relevant quantum applications. We present RASCqL, a reaction-time-limited architecture for space-time efficient complex-instruction-set quantum computation with qLDPC logic. RASCqL supports key algorithmic subroutines such as quantum arithmetic and state preparation directly within co-designed qLDPC codes, achieving 2× to 7× reductions in qubit footprint while maintaining space-time volume comparable to state-of-the-art transversal surface-code architectures. Unlike prior approaches that aim for versatile logical instruction sets for arbitrary circuits, RASCqL adopts an application-tailored code modification that embeds specific complex Clifford transformations useful for common subroutines as virtually implementable operations arising from code automorphisms. RASCqL further leverages parallel physical operations available in reconfigurable neutral-atom arrays to enable fast QEC cycles and high-fidelity transversal operations. At the cost of increased design complexity and specialization, RASCqL can improve end-to-end resource estimates for applications such as factoring and quantum chemistry simulation in both footprint and space-time volume under realistic physical error rates of approximately 2×10−3 to 5×10−4, without requiring additional hardware capabilities. These results demonstrate that qLDPC codes can serve as complex quantum logic units for useful quantum algorithms, extending their practical utility in fault-tolerant quantum computing architectures.