面向量子动力学预测的互补矩阵门控QKAN快速权重编程器

摘要

序列模型必须决定将哪些内容写入记忆以及保留哪些内容。在量子与量子启发的序列学习中，非线性循环更新通常需要重复的电路评估和随时间的顺序反向传播，这使得长上下文的处理成本高昂。基于量子启发Kolmogorov-Arnold网络（QKAN）的门控快速权重编程器（FWP）通过将上下文存储在时变快速参数中来缓解这一瓶颈。然而，其标量门控对每个快速状态坐标施加单一的保留-写入平衡，迫使所有参数共享一个记忆时间尺度。我们引入了基于QKAN的自调制FWP，用低秩生成的逐元素调制取代这种广播门控，调制对象可以是新提议分支、有界旧状态分支，或两者。我们进一步提出互补矩阵门控（CMG），它使用一个sigmoid矩阵门来保留旧状态，并用其互补部分写入新提议。CMG提供逐坐标记忆控制，同时保持标量门控的有界凸更新和仿射前缀扫描结构，其代价仅为单分支规则的调制头。我们在四种FWP架构上比较了四种自调制规则与标量门控，这些架构结合了经典以及基于QKAN的慢速和快速编程器。在七个单步预测基准和五种序列长度上，对于快速编程器包含QKAN模块的架构，CMG提供了最一致的改进。在使用CUDA-Q Dynamics模拟的Jaynes-Cummings和transmon-谐振器动力学的直接多步预测中，CMG模型在4、8和16步的预测区间上保持均方误差在0.001或更低量级，同时其均方误差相较标量门控对应模型至少降低91.2%。这些结果确立了逐坐标互补调制作为基于QKAN的FWP的一种稳定且有效的更新方式。

English

Sequence models must decide what to write into memory and what to retain. In quantum and quantum-inspired sequence learning, nonlinear recurrent updates often require repeated circuit evaluations and sequential backpropagation through time, making long contexts costly. Gated fast-weight programmers (FWPs) based on quantum-inspired Kolmogorov-Arnold networks (QKANs) alleviate this bottleneck by storing context in time-varying fast parameters. However, their scalar gate applies one retention-write balance to every fast-state coordinate, forcing all parameters to share a memory timescale. We introduce Self-Modulating QKAN-based FWPs, which replace this broadcast gate with low-rank-generated element-wise modulation of the new-proposal branch, a bounded old-state branch, or both. We further propose Complementary Matrix Gating (CMG), which uses one sigmoid matrix gate to retain the old state and its complement to write the new proposal. CMG provides coordinate-wise memory control while preserving the bounded convex update and affine prefix-scan structure of scalar gating, at the modulation-head cost of a single-branch rule. We compare four self-modulating rules with scalar gating across four FWP architectures combining classical and QKAN-based slow and fast programmers. Across seven single-step forecasting benchmarks and five sequence lengths, CMG gives the most consistent improvements for architectures whose fast programmer incorporates a QKAN-based module. In direct multi-step forecasting of Jaynes-Cummings and transmon-resonator dynamics simulated with CUDA-Q Dynamics, CMG models maintain mean-squared errors on the order of 0.001 or lower across forecasting horizons of 4, 8, and 16 steps, while improving on their scalar-gated counterparts by at least 91.2%. These results establish coordinate-wise complementary modulation as a stable and effective update for QKAN-based FWPs.