  Quantum simulations of bosonic field theories require a truncation in field
space to map the theory onto finite quantum registers. Ideally, the truncated
theory preserves the symmetries of the original model and has a critical point
in the same universality class. In this paper, we explore two different
truncations that preserve the symmetries of the 1+1-dimensional $O(3)$
non-linear $\sigma$-model - one that truncates the Hilbert space for the unit
sphere by setting an angular momentum cutoff and a fuzzy sphere truncation
inspired by non-commutative geometry. We compare the spectrum of the truncated
theories in a finite box with the full theory. We use open boundary conditions,
a novel method that improves on the correlation lengths accessible in our
calculations. We provide evidence that the angular-momentum truncation fails to
reproduce the $\sigma$-model and that the anti-ferromagnetic fuzzy model agrees
with the full theory.
