Bosonic quantum field theories, even when regularized using a finite lattice, possess an infinite dimensional Hilbert space and, therefore, cannot be simulated in quantum computers with a finite number of qubits. A truncation of the Hilbert space is then needed and the physical results are obtained after a double limit: one to remove the truncation and another to remove the regulator (the continuum limit). A simpler alternative is to find a model with a finite dimensional Hilbert space belonging to the same universality class as the continuum model (a qubitization), so only the space continuum limit is required. A qubitization of the $1+1$ dimensional asymptotically free $O(3)$ nonlinear $sigma$-model based on ideas of non-commutative geometry was previously proposed arXiv:1903.06577 and, in this paper, we provide evidence that it reproduces the physics of the $sigma$-model both in the infrared and the ultraviolet regimes.
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