Complementary Matrix-Gated QKAN Fast-Weight Programmers for Quantum Dynamics Forecasting
Kuo-Chung Peng, Samuel Yen-Chi Chen, Jiun-Cheng Jiang, Chen-Yu Liu, En-Jui Kuo, Yun-Yuan Wang, Tzung-Chi Huang, Prayag Tiwari, Chi-Sheng Chen, Chun-Hua Lin, Yu-Chao Hsu, Tai-Yue Li, Saif Al-Kuwari, Simon See, Kuan-Cheng Chen, Nan-Yow Chen, Hsi-Sheng Goan
quant-ph, cs.AI, cs.LG
2026-07-30
A complementary matrix gate gives QKAN fast-weight programmers per-coordinate memory control without losing prefix-scan parallelism, cutting forecast error over 91%.
This is an architecture paper for "quantum-inspired" sequence models, not a mainstream LLM problem. It targets a memory mechanism called a fast-weight programmer (FWP): a slow network "programs" a separate set of time-varying fast parameters that act as memory. QKAN-based FWPs, built on quantum-inspired Kolmogorov-Arnold networks, store context in those fast parameters and sidestep the cost of repeatedly evaluating quantum circuits and backpropagating step by step through time in nonlinear recurrent updates.
The flaw in existing gated QKAN-FWPs is the gate itself: a single scalar that broadcasts one retention-versus-write balance to every coordinate of the fast state. Every parameter is then forced to share a single memory timescale, with no way to let some dimensions remember the long term and others the short term.
Two changes. First, Self-Modulating QKAN-FWPs: the slow programmer generates low-rank modulation matrices that act element-wise on the new-proposal branch, the bounded old-state branch, or both, replacing the scalar broadcast. The paper compares four rules: Only-new (modulate the proposal), Only-old (modulate retention, tanh-bounded), Full (both), and CMG.
Second, the headline contribution, Complementary Matrix Gating (CMG): a single low-rank-generated sigmoid matrix gate G retains the old state while its complement 1-G writes the new proposal:
Thetat = Gt (mult) Theta{t-1} + (1 - Gt) (mult) Deltat
Because G is a matrix, each coordinate gets its own retention/write balance and thus its own memory timescale. The update stays a bounded convex combination of old state and new proposal and keeps an affine structure, so it remains compatible with a prefix scan and needs no step-by-step backpropagation through time. That is the selling point: per-coordinate control without giving up parallelism, at the modulation-head cost of a single-branch rule.
The authors run four self-modulating rules against scalar gating across seven single-step forecasting benchmarks and four sequence lengths. Over all 28 configurations (7 by 4), CMG variants achieve the lowest MSE in 24. The gains are most consistent in architectures where a QKAN module sits inside the fast programmer.
The load-bearing test is multi-step forecasting. Using CUDA-Q Dynamics to simulate Jaynes-Cummings and transmon-resonator quantum dynamics, CMG models forecast 4, 8, and 16 steps with MSE no worse than 9.3e-4 across all horizons, improving on their scalar-gated counterparts by at least 91.2%. On transmon-resonator the improvement reaches 99.9% to 100%. By contrast the Only-new rule fails badly, degrading performance by up to 223%. On parameter efficiency, CMG needs 139.3% of the gated parameter count for QKAN-QKANFWP, versus 182.1% for Full self-modulation.
For anyone working on quantum-inspired sequence models or fast-weight programmers, CMG is a clean, cheap upgrade: swap the scalar gate for a low-rank matrix gate, get per-coordinate memory control, and keep both prefix-scan parallelism and the stable bounded-convex update. That is its real value range.
The honest caveat is that this audience is narrow. Quantum-inspired FWPs are far from the mainstream of sequence modeling, and every benchmark here is a quantum-dynamics (physics simulation) task. For people doing general time-series or language modeling, the direct takeaways are limited.
The authors acknowledge that the current formulation targets univariate temporal memory and does not model spatial structure among variables; structured spatio-temporal fast states are their named next step. One concern on close reading: among the four modulation rules, Only-new is clearly unstable on multi-step forecasting (up to 223% worse), which suggests element-wise modulation is fragile and the gains of CMG are tied to its specific complementary-convex form. The paper's ablation of why this particular parameterization stays stable is thin.