Constraint-Optimal Driven Allocation for Scalable QEC Decoder Scheduling

TL;DR

The paper tackles the bottleneck of decoding resource scarcity in large-scale FTQC by introducing CODA, an optimization-based decoder scheduler that leverages global circuit structure to minimize the longest undecoded sequence. By reformulating the problem as a sequence of time-bounded feasibility checks solved with CP-SAT, CODA achieves substantial backlog reductions (average 74% across 19 benchmarks) while maintaining linear scalability with circuit size. Compared with heuristic policies, CODA consistently delivers the shortest undecoded sequences and demonstrates practical runtime up to hundreds of qubits, addressing both performance and scalability for decoder virtualization in FTQC. This approach provides a robust, globally optimized scheduling framework that enables efficient use of limited decoders in future large-scale quantum systems.

Abstract

Fault-tolerant quantum computing (FTQC) requires fast and accurate decoding of Quantum Error Correction (QEC) syndromes. However, in large-scale systems, the number of available decoders is much smaller than the number of logical qubits, leading to a fundamental resource shortage. To address this limitation, Virtualized Quantum Decoder (VQD) architectures have been proposed to share a limited pool of decoders across multiple qubits. While the Minimize Longest Undecoded Sequence (MLS) heuristic has been introduced as an effective scheduling policy within the VQD framework, its locally greedy decision-making structure limits its ability to consider global circuit structure, causing inefficiencies in resource balancing and limited scalability. In this work, we propose Constraint-Optimal Driven Allocation (CODA), an optimization-based scheduling algorithm that leverages global circuit structure to minimize the longest undecoded sequence length. Across 19 benchmark circuits, CODA achieves an average 74\% reduction in the longest undecoded sequence length. Crucially, while the theoretical search space scales exponentially with circuit size, CODA effectively bypasses this combinatorial explosion. Our evaluation confirms that the scheduling time scales linearly with the number of qubits, determined by physical resource constraints rather than the combinatorial search space, ensuring robust scalability for large-scale FTQC systems. These results demonstrate that CODA provides a global optimization-based, scalable scheduling solution that enables efficient decoder virtualization in large-scale FTQC systems.

Figures (7)

• Figure 1: Resource imbalance in large-scale FTQC system. Here, the scalability of decoder resources is limited by practical constraints such as hardware cost and power consumption, resulting in $m < n$ and creating a resource imbalance between available decoders and logical qubits.
• Figure 2: Examples of decoder scheduling under different scenarios. (a) Ideal case, where the number of decoders matches the number of logical qubits. Each logical qubit $Q_i$ is continuously assigned to its dedicated decoder $D_i$ across all time slices, resulting in no idle states and no accumulation of undecoded data. (b) Resource-limited case, where the number of available decoders is smaller than the number of logical qubits. Some qubits remain idle in certain time slices, and the same decoder is reused across multiple qubits over time.
• Figure 3: Overall CODA-driven QEC decoder scheduler workflow. It takes two inputs: syndrome data from the syndrome buffer and decoder data from the decoder pool. The scheduler consists of two main components: (1) Constraint Generator that defines resource constraints (the number of available decoders) and performance constraints (allocatable position), and (2) Optimization Solver that determines the optimal schedule via the objective function (minimizing undecoded sequence length), time-bounded search (limiting computation time), and solution selection (identifying the minimum feasible gap value). After a series of processes, the scheduler returns the map that assigns decoders to logical qubits.
• Figure 4: Scheduling example comparing CODA with MLS for 6 logical qubits and 3 available decoders. MLS prioritizes the qubit with the longest undecoded sequence at each time slice but cannot anticipate mandatory decoder allocations required by future $T$-gate operations, leading to extended undecoded sequences such as $5t_{seq}$ for $Q_4$. In contrast, CODA reduces the longest undecoded sequence length to $3t_{seq}$ by employing a global optimization approach that accounts for future precedence constraints.
• Figure 5: Comparison of the longest undecoded sequence length obtained from RR, MLS, and the proposed CODA scheduling policy over a range of benchmark circuits, as well as their geometric mean (g-mean). The results demonstrate that CODA consistently shows the shortest undecoded sequences among all benchmarks, with precise numerical values reported in Table \ref{['tab:benchmark_results']}.