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Pulsar timing arrays (PTAs) detect low-frequency gravitational waves (GWs) by looking for correlated deviations in pulse arrival times. Current Bayesian searches use Markov Chain Monte Carlo (MCMC) methods, which struggle to sample the large number of parameters needed to model the PTA and GW signals. As the data span and number of pulsars increase, this problem will only worsen. An alternative Monte Carlo sampling method, Hamiltonian Monte Carlo (HMC), utilizes Hamiltonian dynamics to produce sample proposals informed by first-order gradients of the model likelihood. This in turn allows it to converge faster to high dimensional distributions. We implement HMC as an alternative sampling method in our search for an isotropic stochastic GW background, and show that this method produces equivalent statistical results to similar analyses run with standard MCMC techniques, while requiring 100-200 times fewer samples. We show that the speed of HMC sample generation scales as $\mathcal{O}(N_\mathrm{psr}^{5/4})$ where $N_\mathrm{psr}$ is the number of pulsars, compared to $\mathcal{O}(N_\mathrm{psr}^2)$ for MCMC methods. These factors offset the increased time required to generate a sample using HMC, demonstrating the value of adopting HMC techniques for PTAs.