  We determine the mass gap of a two-dimensional $O(3)$ nonlinear sigma model
augmented with a topological $\theta$-term using tensor network and digital
quantum algorithms. As proof of principle, we consider the example $\theta =
\pi$ and study its critical behaviour on a quantum simulator by examining the
entanglement entropy of the ground state. We confirm that the quantum theory is
massless in the strong-coupling regime, in agreement with analytical results.
However, we also highlight the limitations of current quantum algorithms,
designed for noisy intermediate-scale quantum devices, in the theory simulation
at weak coupling. Finally, we compare the performance of our quantum algorithms
to classical tensor network methods.
