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From asymptotic freedom to $θ$ vacua: Qubit embeddings of the O(3) nonlinear $σ$ model

Conventional lattice formulations of $θ$ vacua in the $1+1$-dimensional $\text{O}(3)$ nonlinear sigma model suffer from a sign problem. Here, we construct the first sign-problem-free regularization for arbitrary $θ$. Using efficient lattice Monte Carlo algorithms, we demonstrate how a Hamiltonian model of spin-$\tfrac12$ degrees of freedom on a 2-dimensional spatial lattice reproduces both the infrared sector for arbitrary $θ$, as well as the ultraviolet physics of asymptotic freedom. Furthermore, as a model of qubits on a two-dimensional square lattice with only nearest-neighbor interactions, it is naturally suited for studying the physics of $θ$ vacua and asymptotic freedom on near-term quantum devices. Our construction generalizes to $θ$ vacua in all $\text{CP}(N-1)$ models, solving a long standing sign problem.