Engineering entanglement geometry via spacetime-modulated measurements

We introduce a general approach to realize quantum states with holographic entanglement structure via monitored dynamics. Starting from random unitary circuits in $1+1$ dimensions, we introduce measurements with a spatiotemporally modulated density. Exploiting the known critical properties of the measurement-induced entanglement transition, this allows us to engineer arbitrary geometries for the bulk space (with a fixed topology). These geometries in turn control the entanglement structure of the boundary (output) state. We demonstrate our approach by giving concrete protocols for two geometries of interest in two dimensions: the hyperbolic half-plane and a spatial section of the Ba\~nados-Teitelboim-Zanelli black hole. We numerically verify signatures of the underlying entanglement geometry, including a direct imaging of entanglement wedges by using locally entangled reference qubits. Our results provide a concrete platform for realizing geometric entanglement structures on near-term quantum simulators.