用於量子動力學預測的互補矩陣門控QKAN快速權重編程器

摘要

序列模型必須決定要將什麼寫入記憶體，以及保留什麼。在量子與量子啟發的序列學習中，非線性循環更新通常需要重複的電路評估與隨時間的順序反向傳播，使得長上下文處理成本高昂。基於量子啟發柯爾莫哥洛夫-阿諾德網路（QKANs）的閘控快權重程式器（FWPs）通過將上下文儲存於隨時間變化的快參數中，緩解了此瓶頸。然而，其純量閘控將單一的保留-寫入平衡應用於每個快狀態座標，迫使所有參數共享同一記憶時間尺度。我們引入了自調製的基於QKAN的FWPs，以低秩生成的逐元素調製取代此廣播閘控，作用於新提案分支、有界舊狀態分支，或兩者。我們進一步提出互補矩陣閘控（CMG），其使用一個sigmoid矩陣閘控來保留舊狀態，並以其補數寫入新提案。CMG提供逐座標記憶控制，同時保留了純量閘控的有界凸組合更新與仿射前綴掃描結構，僅需單分支規則的調製頭成本。我們在結合經典與QKAN基礎的慢速與快速程式器的四種FWP架構中，比較了四種自調製規則與純量閘控。在七個單步預測基準與五種序列長度中，CMG對其快速程式器包含QKAN基礎模組的架構提供了最一致的改進。在利用CUDA-Q Dynamics模擬的Jaynes-Cummings與transmon-resonator動力學的直接多步預測中，CMG模型在4、8和16步的預測範圍內維持均方誤差在0.001或更低之數量級，同時比其對應的純量閘控版本至少改善91.2%。這些結果確立了逐座標互補調製作為基於QKAN的FWPs之一種穩定且有效的更新機制。

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.