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Quantum simulations would be highly desirable in order to investigate thefinite density physics of QCD. $(1+1)$-d $\mathbb{C}P(N-1)$ quantum fieldtheories are toy models that share many important features of QCD: they areasymptotically free, have a non-perturbatively generated massgap, as well as$\theta$-vacua. $SU(N)$ quantum spin ladders provide an unconventionalregularization of $\mathbb{C}P(N-1)$ models that is well-suited for quantumsimulation with ultracold alkaline-earth atoms in an optical lattice. In orderto validate future quantum simulation experiments of $\mathbb{C}P(2)$ models atfinite density, here we use quantum Monte Carlo simulations on classicalcomputers to investigate $SU(3)$ quantum spin ladders at non-zero chemicalpotential. This reveals a rich phase structure, with single- or double-speciesBose-Einstein "condensates", with or without ferromagnetic order.

SU(3) Quantum Spin Ladders as a Regularization of the CP(2) Model at Non-Zero Density: From Classical to Quantum Simulation