Given the currently available noisy Quantum Hardware, a reliable and fast Quantum Error Correction is required to build functional Quantum Computers. Based on the Surface Code-arrangement of Qubits, several different error decoders have been proposed to solve the underlying problem of Minimum Weight Perfect Matching, such as Union-Find on Systolic-Array-like distributed computing architectures. Since these published approaches have a relatively large hardware footprint, they show a lack of scalability for the required large code distances in the Surface Code. We therefore propose an implementation of Union-Find on a multi-context Systolic Array that allows for efficient and latency-optimized Quantum Error Correction of large Surface Codes with currently available hardware through time-multiplexing.
Given the currently available noisy Quantum Hardware, a reliable and fast Quantum Error Correction is required to build functional Quantum Computers. Based on the Surface Code-arrangement of Qubits, several different error decoders have been proposed to solve the underlying problem of Minimum Weight Perfect Matching, such as Union-Find on Systolic-Array-like distributed computing architectures. Since these published approaches have a relatively large hardware footprint, they show a lack of scalability for the required large code distances in the Surface Code. We therefore propose an implementation of Union-Find on a multi-context Systolic Array that allows for efficient and latency-optimized Quantum Error Correction of large Surface Codes with currently available hardware through time-multiplexing.
Quantum Error Correction is an essential key ingredient for large-scale Quantum Computing devices. Indeed, a fast, scalable, and accurate decoding and correction process of errors appearing in qubits is critical to guarantee fault tolerance and usability. In this context, the Union-Find decoder incarnates a promising approach for such an essential task, as proved by various literature studies. Nonetheless, further advancements in this area are still feasible. For this reason, we propose a Systolic Array-based architecture for the Union-Find decoder that exploits a novel communication and decoding scheme to speed up the overall process. Furthermore, we design three solutions based on such a scheme and evaluate them according to different metrics, such as latency and accuracy.
Reliable large-scale quantum computation relies on fault-tolerant architectures, where quantum error correction (QEC) continuously extracts and decodes error syndromes in real time. A critical component in QEC is the decoder, a classical subsystem that must simultaneously deliver high logical accuracy and ultra-low latency. This paper presents a novel algorithm-hardware co-design that improves the accuracy-latency trade-off over existing approaches such as vanilla Minimum-Weight Perfect Matching (MWPM) and Union-Find (UF) decoders. At the algorithmic level, we introduce coset ensemble decoding, which improves UF decoding by explicitly exploiting logically equivalent cosets. Our method performs ensemble forest exploration to generate multiple coset-consistent candidates and aggregates them to approximate coset-level maximum-likelihood decoding. We further reduce computational and memory complexity via reverse-order elimination and lossless graph compression, without sacrificing accuracy. At the hardware level, we design a domain-specific architecture that temporally reuses resources, avoiding the code-distance-proportional resource growth in prior spatial architectures. Several optimizations, such as multi-bank memory hashing and hierarchical ID mapping, are proposed to mitigate pipeline stalls and memory conflicts under highly concurrent access patterns. Under a circuit-level depolarizing noise model, our co-design approach achieves a better accuracy-latency trade-off than prior MWPM- and UF-based decoders, while reducing FPGA LUT consumption by up to 8.2 times compared with reported UF-based decoder resources. The tunable candidate number further exposes a flexible design knob, enabling users to tailor decoding performance to the requirements of different fault-tolerant workloads. Our implementation is publicly available at https://github.com/IMSeonL/coset-ensemble-decoder.
The union-find decoder is a leading algorithmic approach to the correction of quantum errors on the surface code, achieving code thresholds comparable to minimum-weight perfect matching (MWPM) with amortized computational time scaling near-linearly in the number of physical qubits. This complexity is achieved via optimizations provided by the disjoint-set data structure. We demonstrate, however, that the behavior of the decoder at scale underutilizes this data structure for twofold analytic and algorithmic reasons, and that improvements and simplifications can be made to architectural designs to reduce resource overhead in practice. To reinforce this, we model the behavior of erasure clusters formed by the decoder and show that there does not exist a percolation threshold within the data structure for any mode of operation. This yields a linear-time worst-case complexity for the decoder at scale, even with a naive implementation omitting popular optimizations. Published by the American Physical Society 2024
Quantum computation promises significant computational advantages over classical computation for some problems. However, quantum hardware suffers from much higher error rates than in classical hardware. As a result, extensive quantum error correction is required to execute a useful quantum algorithm. The decoder is a key component of the error correction scheme whose role is to identify errors faster than they accumulate in the quantum computer and that must be implemented with minimum hardware resources in order to scale to the regime of practical applications. In this work, we consider surface code error correction, which is the most popular family of error correcting codes for quantum computing, and we design a decoder micro-architecture for the Union-Find decoding algorithm. We propose a three-stage fully pipelined hardware implementation of the decoder that significantly speeds up the decoder. Then, we optimize the amount of decoding hardware required to perform error correction simultaneously over all the logical qubits of the quantum computer. By sharing resources between logical qubits, we obtain a 67% reduction of the number of hardware units and the memory capacity is reduced by 70%. Moreover, we reduce the bandwidth required for the decoding process by a factor at least 30x using low-overhead compression algorithms. Finally, we provide numerical evidence that our optimized micro-architecture can be executed fast enough to correct errors in a quantum computer.
Quantum error correction is one of the most important milestones for realization of large-scale quantum computation. To achieve this, it is essential not only to integrate a large number of qubits with high fidelity, but also to build a scalable classical system that can perform error correction. Here, we propose an efficient and scalable decoder for quantum error correction using Fujitsu Digital Annealer (DA). Specifically, the error correction problem of stabilizer codes is mapped into an Ising-type optimization problem, so-called quadratic unconstrained binary optimization (QUBO) problem, which is solved by DA. In particular, we implement the proposed DA decoder for the surface code and perform detailed numerical experiments for various code distances to see its performance and scalability. We observe that computational scaling for the DA decoder has a lower order of polynomial than the decoding methods using simulated annealing (SA) and minimum-weight perfect matching (MWPM) algorithm under all tested conditions. It is also shown that the DA decoder has advantages over the Union-Find (UF) decoder from a variety of perspectives including hardware implementation. Furthermore, the threshold behavior of the logical error probability for the DA decoder is analyzed and the resultant threshold lies between 9.4% and 9.8%, which is very close to that obtained by the MWPM decoder. This result clearly shows the high potential of the DA decoder for quantum error correction.
Quantum error correction is one of the most important milestones for the realization of large-scale quantum computation. To achieve this, it is essential not only to integrate a large number of qubits with high fidelity but also to build a scalable classical system that can perform error correction. Here, we propose an efficient and scalable decoder for quantum error correction using ``Fujitsu Digital Annealer'' (DA). Specifically, the error correction problem of stabilizer codes is mapped into an Ising-type optimization problem, so-called quadratic unconstrained binary optimization (QUBO) problem, which is solved by DA. In particular, we implement the proposed DA decoder for the surface code and perform detailed numerical experiments for various code distances to see its performance and scalability. We observe that computational scaling for the DA decoder has a lower order of polynomials than the decoding methods using simulated annealing (SA) and minimum-weight perfect matching (MWPM) algorithm under all tested conditions. It is also shown that the DA decoder has advantages over the Union-Find (UF) decoder from a variety of perspectives including hardware implementation. Furthermore, the threshold behavior of the logical error probability for the DA decoder is analyzed and the resultant threshold lies between 9.4% and 9.8%, which is very close to that obtained by the MWPM decoder. This result clearly shows the high potential of the DA decoder for quantum error correction.