Frustration-Free Control and Absorbing-State Transport in Entangled State Preparation

We introduce frustration-free control, a measurement-feedback framework for quantum state preparation that extends frustration-free Hamiltonians to monitored stochastic dynamics. The protocol drives many-body systems into highly entangled target states, common dark states of all measurement projectors, through minimal local unitary corrections that realize an absorbing-state dynamics without postselection. We show that relaxation to the target state is governed by emergent transport of nonlocal charges, such as singlet excitations in SU(2)-symmetric dynamics. While measurement feedback annihilates compatible charge configurations, both measurement and scrambling unitaries induce charge transport and thus determine the convergence time. Mapping a baseline model of SU(N) SWAP measurements with local corrections to a solvable absorbing random walk yields a run-time scaling t∼L^{z} with transport exponent z=2. Simulations of Motzkin and Fredkin chains reveal subdiffusive scaling z≥8/3, confirming absorbing-state transport as a general organizing principle for deterministic entangled-state preparation and transport diagnostics in monitored quantum dynamics.