Hamiltonian Monte Carlo from $q$-deformed phase-space mechanics

Abstract

Hamiltonian Monte Carlo (HMC) generates efficient Markov transitions by combining Hamiltonian dynamics with a Metropolis correction. This paper develops a geometric \(q\)-analogue of HMC by replacing classical Hamiltonian dynamics with a \(q\)-deformed Hamiltonian system arising from \(q\)-calculus. Starting from a Lagrangian formulation, we derive the corresponding \(q\)-Hamiltonian equations and prove the formal invariance of the associated \(q\)-symplectic form within the \(q\)-deformed differential calculus. To obtain a computable sampler, we introduce a Jackson-derivative realization and construct a Metropolis-corrected \(q\)-HMC algorithm. The proposal reduces to classical HMC as \(q\to1\), while for \(q\neq1\) it replaces ordinary derivatives by \(q\)-Jackson finite differences. We establish detailed balance, which ensures that the resulting Markov transition preserves the target distribution. Numerical experiments examine the computational behavior of the proposed method. For positive-scale black-box targets, the \(q\)-Jackson force has a scale-consistent interpretation: multiplicative perturbations of \(s>0\) correspond to centered finite differences in \(y=\log s\). In such examples, \(q\)-HMC closely tracks log-coordinate finite-difference HMC and the exact-gradient benchmark, whereas raw additive finite differences may produce large force and Hamiltonian errors. These results suggest that the proposed \(q\)-analogue provides a valid HMC-type sampling framework with a visible advantage for positive and multiplicative black-box targets.