  The ability to selectively measure, initialize, and reuse qubits during a
quantum circuit enables a mapping of the spatial structure of certain
tensor-network states onto the dynamics of quantum circuits, thereby achieving
dramatic resource savings when using a quantum computer to simulate many-body
systems with limited entanglement. We experimentally demonstrate a significant
benefit of this approach to quantum simulation: In addition to all correlation
functions, the entanglement structure of an infinite system -- specifically the
half-chain entanglement spectrum -- is conveniently encoded within a small
register of "bond qubits" and can be extracted with relative ease. Using a
trapped-ion QCCD quantum computer equipped with selective mid-circuit
measurement and reset, we quantitatively determine the near-critical
entanglement entropy of a correlated spin chain directly in the thermodynamic
limit and show that its phase transition becomes quickly resolved upon
expanding the bond-qubit register.
