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

We study frustration-free control, a measurement-feedback protocol for quantum state preparation that extends the concept of frustration-free Hamiltonians to 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 post-selection. 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 runtime scaling t sim L^z with transport exponent z=2. Simulations of Motzkin and Fredkin chains reveal subdiffusive scaling z ge tfrac{8}{3}, confirming the transport picture and suggesting strategies for controlled entangled-state preparation and charge-transport probing in monitored quantum dynamics.