The realization of scalable fault-tolerant quantum computing is expected to hinge on quantum error-correcting codes. In the quest for more efficient quantum fault tolerance, a critical code parameter is the weight of measurements that extract information about errors to enable error correction: as higher measurement weights require higher implementation costs and introduce more errors, it is important in code design to optimize measurement weight. This underlies the surging interest in quantum low-density parity-check (qLDPC) codes, the study of which has primarily focused on the asymptotic (large-code-limit) properties. In this work, we introduce a versatile and computationally efficient approach to stabilizer code weight reduction based on reinforcement learning (RL), which produces new low-weight codes that substantially outperform the state of the art in practically relevant parameter regimes, extending significantly beyond previously accessible small distances. For example, our approach demonstrates savings in physical qubit overhead compared to existing results by 1 to 2 orders of magnitude for weight 6 codes and brings the overhead into a feasible range for near-future experiments. We also investigate the interplay between code parameters using our RL framework, offering new insights into the potential efficiency and power of practically viable coding strategies. Overall, our results demonstrate how RL can effectively advance the crucial yet challenging problem of quantum code discovery and thereby facilitate a faster path to the practical implementation of fault-tolerant quantum technologies.
The realization of scalable fault-tolerant quantum computing is expected to hinge on quantum error-correcting codes. In the quest for more efficient quantum fault tolerance, a critical code parameter is the weight of measurements that extract information about errors to enable error correction: as higher measurement weights require higher implementation costs and introduce more errors, it is important in code design to optimize measurement weight. This underlies the surging interest in quantum low-density parity-check (qLDPC) codes, the study of which has primarily focused on the asymptotic (large-code-limit) properties. In this work, we introduce a versatile and computationally efficient approach to stabilizer code weight reduction based on reinforcement learning (RL), which produces new low-weight codes that substantially outperform the state of the art in practically relevant parameter regimes, extending significantly beyond previously accessible small distances. For example, our approach demonstrates savings in physical qubit overhead compared to existing results by 1 to 2 orders of magnitude for weight 6 codes and brings the overhead into a feasible range for near-future experiments. We also investigate the interplay between code parameters using our RL framework, offering new insights into the potential efficiency and power of practically viable coding strategies. Overall, our results demonstrate how RL can effectively advance the crucial yet challenging problem of quantum code discovery and thereby facilitate a faster path to the practical implementation of fault-tolerant quantum technologies.
Low check weight is a crucial code property for fault-tolerant quantum computing, which underlies the strong interest in quantum low-density parity-check (qLDPC) codes. Here, we initiate the theory of weight-constrained stabilizer codes from various foundational perspectives including the complexity of computing code weight and the explicit boundary of feasible low-weight codes in both theoretical and practical settings. We first prove that computing the optimal generator weight of a stabilizer code is NP-hard, motivating efficiently computable bounds. We derive analytical lower bounds on check weight in terms of code rate and distance, identifying the minimum weights needed for single-qubit error detection and correction, as well as the sharp distance and rate limits of weight-three error-detecting codes. Matching constructions show that these bounds are tight in several regimes. To establish refined finite-size constraints, we develop a linear programming framework based on quantum weight enumerators subject to generator-weight constraints, yielding exact optimal weights for all parameter combinations with n≤9. Finally, we show that the same framework can incorporate architecture-dependent constraints, using the 127-qubit IBM Eagle chip as a concrete example. Our study brings the weight as a crucial parameter into coding theory and provides guidance for code design and utility in practical scenarios.
The realization of fault-tolerant quantum computing relies fundamentally on the efficient and accurate decoding of quantum error-correcting codes. Recently, Deep Reinforcement Learning (DRL) has emerged as a promising approach for decoding complex codes, such as quantum Low-Density Parity-Check (qLDPC) codes, due to its ability to adapt to correlated noise and system specific irregularities. However, the computational cost of training these intelligent agents scales prohibitively with the code distance and the number of physical qubits. In this work, we present an analytical and theoretical framework for transfer learning in DRL-based quantum decoders. We explore the scale-invariant properties of qLDPC codes, specifically focusing on how the structural symmetries in hypergraph product and lifted product codes can be exploited by scalable neural architectures. By formulating the decoding process as a Markov Decision Process (MDP) mapped onto a translationally and structurally invariant graph, we demonstrate that a Deep Q-Network (DQN) trained on a small lattice (e.g., an [[625,25]] qLDPC code) can be zero-shot transferred or rapidly fine-tuned for a significantly larger lattice (e.g., an [[900,50]] qLDPC code). We provide rigorous theoretical bounds on the performance degradation under dimensionality mismatch and present comprehensive empirical results. Our experiments show that fine-tuning a pre-trained agent requires orders of magnitude fewer samples than training from scratch, while retaining near-optimal logical error suppression. These findings provide a concrete pathway toward scalable, low-latency machine learning architectures for large-scale fault-tolerant quantum processors.
The realization of fault-tolerant quantum computing relies fundamentally on the efficient and accurate decoding of quantum error-correcting codes. Recently, Deep Reinforcement Learning (DRL) has emerged as a promising approach for decoding complex codes, such as quantum Low-Density Parity-Check (qLDPC) codes, due to its ability to adapt to correlated noise and system specific irregularities. However, the computational cost of training these intelligent agents scales prohibitively with the code distance and the number of physical qubits. In this work, we present an analytical and theoretical framework for transfer learning in DRL-based quantum decoders. We explore the scale-invariant properties of qLDPC codes, specifically focusing on how the structural symmetries in hypergraph product and lifted product codes can be exploited by scalable neural architectures. By formulating the decoding process as a Markov Decision Process (MDP) mapped onto a translationally and structurally invariant graph, we demonstrate that a Deep Q-Network (DQN) trained on a small lattice (e.g., an [[625,25]] qLDPC code) can be zero-shot transferred or rapidly fine-tuned for a significantly larger lattice (e.g., an [[900,50]] qLDPC code). We provide rigorous theoretical bounds on the performance degradation under dimensionality mismatch and present comprehensive empirical results. Our experiments show that fine-tuning a pre-trained agent requires orders of magnitude fewer samples than training from scratch, while retaining near-optimal logical error suppression. These findings provide a concrete pathway toward scalable, low-latency machine learning architectures for large-scale fault-tolerant quantum processors.
Quantum error correction (QEC) is essential for scalable quantum computing. However, it requires classical decoders that are fast and accurate enough to keep pace with quantum hardware. While quantum low-density parity-check codes have recently emerged as a promising route to efficient fault tolerance, current decoding algorithms do not allow one to realize the full potential of these codes in practical settings. Here, we introduce a convolutional neural network decoder that exploits the geometric structure of QEC codes, and use it to probe a novel "waterfall" regime of error suppression, demonstrating that the logical error rates required for large-scale fault-tolerant algorithms are attainable with modest code sizes at current physical error rates, and with latencies within the real-time budgets of several leading hardware platforms. For example, for the [144,12,12] Gross code, the decoder achieves logical error rates up to ∼17x below existing decoders - reaching logical error rates ∼10−10 at physical error p=0.1% - with 3-5 orders of magnitude higher throughput. This decoder also produces well-calibrated confidence estimates that can significantly reduce the time overhead of repeat-until-success protocols. Taken together, these results suggest that the space-time costs associated with fault-tolerant quantum computation may be significantly lower than previously anticipated.
The realization of large-scale quantum computers requires not only quantum error correction but also fault-tolerant (FT) operations to handle errors that propagate into harmful errors. Recently, flag-based protocols have been introduced that use ancillary qubits to flag harmful errors. However, there is no clear recipe for finding a FT quantum circuit with flag-based protocols, especially when we consider hardware constraints, such as the qubit connectivity and available gate set. This work presents a novel approach to automatically discover compact and hardware-adapted FT quantum circuits to make significant progress towards scalable FT quantum computing. We employ reinforcement learning (RL) as an enabling tool, leveraging a fast, parallelized stabilizer quantum circuit simulator and a nontrivial reward function specifically adapted to the problem. We show that, in the task of FT logical state preparation, RL discovers not only circuits with fewer gates and ancillary qubits than published results but also novel circuits without and with hardware constraints of up to distance-5 codes with 25 physical qubits, and they can be implemented directly in experiments. Furthermore, RL allows for straightforward exploration of different qubit connectivities and the use of transfer learning to accelerate the discovery. More generally, our work sets the framework towards the use of RL or other machine learning techniques for FT quantum circuit discovery with hardware constraints to make real progress towards the realization of large-scale quantum computers, addressing tasks beyond state preparation, including magic state preparation, logical gate synthesis, and syndrome measurement.
Quantum error correction technology is an important method to eliminate errors during the operation of quantum computers. In order to solve the problem of influence of errors on physical qubits, we propose an approximate error correction scheme that performs dimension mapping operations on surface codes. This error correction scheme utilizes the topological properties of error correction codes to map the surface code dimension to three dimensions. Compared to previous error correction schemes, the present three-dimensional surface code exhibits good scalability due to its higher redundancy and more efficient error correction capabilities. By reducing the number of ancilla qubits required for error correction, this approach achieves savings in measurement space and reduces resource consumption costs. In order to improve the decoding efficiency and solve the problem of the correlation between the surface code stabilizer and the 3D space after dimension mapping, we employ a reinforcement learning (RL) decoder based on deep Q-learning, which enables faster identification of the optimal syndrome and achieves better thresholds through conditional optimization. Compared to the minimum weight perfect matching decoding, the threshold of the RL trained model reaches 0.78%, which is 56% higher and enables large-scale fault-tolerant quantum computation.