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We derive a new constructive procedure to rapidly generate ensembles of phase-covariant dynamical maps that may be associated to the individual spins of a closed quantum system. We do this by first computing the single-spin dynamical maps in small XXZ networks and chains, specialized to the class of initial states that guarantees phase-covariant dynamics for each spin. Since the dynamics in any small, closed system contains oscillatory features associated to the system size, we define an averaging procedure to extract time-homogeneous dynamics. We use the the average map and the set of deviations from the average map in the exactly derived ensembles to motivate the form of distributional functions for map parameters. The distributions then straightforwardly generate arbitrary-sized ensembles of channels, constrained by a few global properties. This procedure can also generate ensembles where individual maps are not phase-covariant although the average map is, corresponding to realizations of disordered, or noisy, Hamiltonians. The construction procedure suggests new ways to realize random families of open-system dynamics, subject to constraints that require the ensemble to approximate a partition of a closed system. 

A<sc>bstract</sc> Building on previous constructions examining how a collection of small, locally interacting quantum systems might emerge via spontaneous symmetry breaking from a single-particle system of high dimension, we consider a larger family of geometric loss functionals and explicitly construct several classes of critical metrics which “know about qubits” (KAQ). The loss functional consists of the Ricci scalar with the addition of the Gauss-Bonnet term, which introduces an order parameter that allows for spontaneous symmetry breaking. The appeal of this method is two-fold: (i) the Ricci scalar has already been shown to have KAQ critical metrics and (ii) exact equations of motions are known for loss functionals with generic curvature terms up to two derivatives. We show that KAQ critical metrics, which are solutions to the equations of motion in the space of left-invariant metrics with fixed determinant, exist for loss functionals that include the Gauss-Bonnet term. We find that exploiting the subalgebra structure leads us to natural classes of KAQ metrics which contain the familiar distributions (GUE, GOE, GSE) for random Hamiltonians. We introduce tools for this analysis that will allow for straightfoward, although numerically intensive, extension to other loss functionals and higher-dimension systems.