The strong interactions of the standard model described by quantum chromodynamics (QCD) pose a challenging problem for classical computation. Emerging quantum platforms provide an exciting possibility for investigating QFTs in previously inaccessible regimes. This has motivated a search for unconventional formulations of QFTs with finite-dimensional local Hilbert spaces, or with “qubit” degrees of freedom. In this talk, I will describe recent work in constructing qubit models for asymptotically-free 1+1d nonlinear sigma models, which are well-known prototypes of QCD. In particular, conventional lattice formulations of topological θ vacua in the 2d O(3) nonlinear sigma model suffer from a severe sign problem on classical computers. Remarkably, by formulating this as model of qubits, not only do we obtain a viable path towards its quantum simulation, but we also obtain the first sign-problem-free regularization for arbitrary θ, solving a longstanding sign problem. We show that such models can reproduce both the IR physics of theta vacua and the UV physics of asymptotic freedom. In the search for new qubit models in higher dimensions and for gauge theories, symmetries and anomalies are a guide. We will discuss how the perspective of recently discovered discrete anomalies provides interesting constraints on possible lattice regularizations towards the goal of finding such qubit models for QCD-like theories.