Quantifying Cluster State Construction with Dipolar Interactions*

Cluster states are quantum entangled states that can serve as resource states for measurement-based quantum computing. Measurement-based quantum computing is equivalent to circuit-based quantum computing but can show advantages if cluster states can be prepared efficiently, i.e., with parallel implementation of entangling gates. Interactions between optically trapped atoms and molecules can be used to engineer cluster states where, for example, the dipole-dipole interaction between polar molecules trapped in optical tweezers entangles molecular rotational states. But parallelization of the entanglement process comes with a trade-off in efficacy because entangling operations between polar molecules will induce cluster state errors from longer-range parts of the dipolar interaction. We construct a measure to diagnose the fidelity of cluster states constructed with errors due to interaction tails. We use the fidelity to maximize parallelizability while minimizing errors and construct procedures using global time-dependent fields to build high-fidelity cluster states with dipole-dipole and other long-range interactions. Our work shows that the construction of cluster states using recent advances in atomic, molecular, and optical physics can be efficiently parallelized in systems with long-range interactions.