In this paper a new Quantum Tunneling Particle Swarm Optimization (QTPSO) algorithm is proposed and applied to the training of feedforward Artificial Neural Networks (ANNs). In the classical Particle Swarm Optimization (PSO) algorithm the value of the cost function at the location of the personal best solution found by each particle cannot increase. This can significantly reduce the explorative ability of the entire swarm. In this paper a new PSO algorithm in which the personal best solution of each particle is allowed to tunnel through hills in the cost function analogous to the Tunneling effect in Quantum Physics is proposed. In quantum tunneling a particle which has insufficient energy to cross a potential barrier can still cross the barrier with a small probability that exponentially decreases with the barrier length. The introduction of the quantum tunneling effect allows particles in the PSO algorithm to escape from local minima thereby increasing the explorative ability of the PSO algorithm and preventing premature convergence to local minima. The proposed algorithm significantly outperforms three state-of-the-art PSO variants on a majority of benchmark neural network training problems.