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Long-range entanglement is an important quantum resource, particularly fortopological orders and quantum error correction. In reality, preparinglong-range entangled states requires a deep unitary circuit, which posessignificant experimental challenges. A promising avenue is offered by replacingsome quantum resources with local operations and classical communication(LOCC). With these classical components, one can communicate outcomes ofmidcircuit measurements in distant subsystems, substantially reducing circuitdepth in many important cases. However, to prepare general long-range entangledstates, finding LOCC-assisted circuits of a short depth remains an openquestion. Here, to address this challenge, we propose a quantum-classicalhybrid algorithm to find optimal LOCC protocols for preparing ground states ofgiven Hamiltonians. In our algorithm, we introduce an efficient way to estimateparameter gradients and use such gradients for variational optimization.Theoretically, we establish the conditions for the absence of barren plateaus,ensuring trainability at a large system size. Numerically, the algorithmaccurately solves the ground state of long-range entangled models, such as theperturbed Greenberger-Horne-Zeilinger state and surface code. Our resultsdemonstrate the advantage of our method over conventional unitary variationalcircuits: the practical advantage in the accuracy of estimated ground-stateenergy and the theoretical advantage in creating long-range entanglement.

Variational LOCC-assisted quantum circuits for long-range entangled states