QM-quantization optimizer (Schrodinger gradient-flow)

Implements quantization-based optimization, a stochastic global optimizer whose quantized objective induces a gradient-flow diffusion that maps onto a Schrödinger equation.

The method quantizes the range of the objective with a monotonically increasing resolution, so the quantization error acts as an annealed noise source. Under the White Noise Hypothesis this turns plain gradient descent into a Langevin-type diffusion whose noise scale shrinks as \(Q_p(t)\uparrow\infty\). Recasting the associated Fokker–Planck equation through the substitution \(\rho = |\psi|^2\) yields a Schrödinger equation, and the resulting tunneling effect is what lets iterates climb out of local minima toward the global optimum.

In continuous time the dynamics are the stochastic differential equation \(dX_t = -\nabla_x f(X_t)\,dt + \sqrt{C_q\,Q_p^{-1}(t)}\,dW_t\). Its Euler–Maruyama discretization gives the per-step parameter update:

\[ \begin{aligned} f_Q &= Q_p^{-1}\left\lfloor Q_p\left(f + \tfrac{1}{2}Q_p^{-1}\right)\right\rfloor, \qquad Q_p(t) = \eta\, b^{\,\bar h(t)} \\ \theta_{t+1} &= \theta_t - \eta\, \nabla_\theta f(\theta_t) + \sqrt{2\,\eta\, Q_p^{-1}(t)}\; \xi_t \end{aligned} \]

where \(\theta\) are the parameters, \(\eta\) the learning rate, \(\nabla_\theta f\) the gradient, \(f_Q\) the quantized objective, \(Q_p(t)\) the monotonically increasing quantization resolution with base \(b\) and power \(\bar h(t)\uparrow\infty\), \(C_q\) the diffusion constant, \(\xi_t \sim \mathcal{N}(0, I)\) the injected noise, and \(\lfloor\cdot\rfloor\) the floor operator.

Reference: Jinwuk Seok, Changsik Cho, "Quantum mechanical framework for quantization-based optimization: from Gradient flow to Schrödinger equation", ICLR 2026 (withdrawn) / arXiv 2026. https://arxiv.org/abs/2603.11536

---

Back to the Canon