QIASO

Implements QIASO (Quantum-Inspired Adaptive Superposition Optimization), a gradient-free optimizer that maintains a probability distribution over a fixed set of candidate weight vectors and reshapes it toward low-loss candidates.

Rather than tracking a single parameter vector, QIASO keeps \(K\) candidates \(\{\theta_1,\dots,\theta_K\}\) together with an amplitude (probability) vector \(p_t\) on the simplex. Each iteration solves a KL-regularized variational problem that trades expected candidate loss against proximity to the previous distribution; its closed-form solution is the exponential-weights (Gibbs / softmax) reweighting of the amplitudes. To escape poor local minima, candidate weights are occasionally perturbed by a quantum-tunneling-inspired Gaussian operator whose firing probability decays exponentially over training.

\[ \begin{aligned} p_{t+1} &= \arg\min_{q \in \Delta_K}\ \Big\{ \langle q, L\rangle + \tfrac{1}{\eta}\, D_{\mathrm{KL}}(q \,\|\, p_t) \Big\}, \\ p_{t+1}(k) &= \frac{p_t(k)\,\exp(-\eta\,\ell_k)}{\sum_{j=1}^{K} p_t(j)\,\exp(-\eta\,\ell_j)}, \qquad \ell_k = L(\theta_k), \\ \theta_k &\leftarrow \theta_k + \epsilon\,\zeta\,, \quad \zeta \sim \mathcal{N}(0,\sigma^2 I), \quad \text{with probability } \rho_t = \rho_0\, e^{-\lambda t}. \end{aligned} \]

where \(\theta_k\) are the candidate weight vectors, \(p_t\) the amplitude (probability) vector on the simplex \(\Delta_K\), \(\ell_k = L(\theta_k)\) the loss of candidate \(k\), \(\eta>0\) the inverse-temperature step parameter, \(D_{\mathrm{KL}}\) the Kullback-Leibler divergence, \(\epsilon\) the tunneling perturbation scale, \(\sigma^2\) its Gaussian variance, and \(\rho_0, \lambda\) the initial perturbation probability and its decay rate.

Reference: Irsa Sajjad, Mashail M. AL Sobhi, "The quantum-inspired adaptive superposition optimization for neural network training", AIMS Mathematics 2026. https://www.aimspress.com/article/doi/10.3934/math.2026010

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