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Conventional lattice formulations of $\theta$ vacua in the $1+1$-dimensional$\text{O}(3)$ nonlinear sigma model suffer from a sign problem. Here, weconstruct the first sign-problem-free regularization for arbitrary $\theta$.Using efficient lattice Monte Carlo algorithms, we demonstrate how aHamiltonian model of spin-$\tfrac12$ degrees of freedom on a 2-dimensionalspatial lattice reproduces both the infrared sector for arbitrary $\theta$, aswell as the ultraviolet physics of asymptotic freedom. Furthermore, as a modelof qubits on a two-dimensional square lattice with only nearest-neighborinteractions, it is naturally suited for studying the physics of $\theta$ vacuaand asymptotic freedom on near-term quantum devices. Our constructiongeneralizes to $\theta$ vacua in all $\text{CP}(N-1)$ models, solving a longstanding sign problem.

From asymptotic freedom to $\theta$ vacua: Qubit embeddings of the O(3) nonlinear $\sigma$ model