A geometric $q$-analogue of Hamiltonian Monte Carlo

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 $q\to 1$, while for $q\mathrm{\ne }1$ 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.