  Controllable quantum many-body systems are platforms for fundamental
investigations into the nature of entanglement and promise to deliver
computational speed-up for a broad class of algorithms and simulations. In
particular, engineering entanglement within a dense spin ensemble can turn it
into a robust quantum memory or a computational platform. Recent experimental
progress in dense central spin systems motivates the design of algorithms that
use a central-spin qubit as a convenient proxy for the ensemble. Here we
propose a protocol that uses a central spin to initialize two dense spin
ensembles into a pure anti-polarized state and from there creates a many-body
entangled state -- a singlet -- from the combined ensemble. We quantify the
protocol performance for multiple material platforms and show that it can be
implemented even in the presence of realistic levels of decoherence. Our
protocol introduces an algorithmic approach to preparation of a known many-body
state and to entanglement engineering in a dense spin ensemble, which can be
extended towards a broad class of collective quantum states.
