Scaling the neutral-atom Rydberg gate quantum computer by collective encoding in holmium atoms

We discuss a method for scaling a neutral-atom Rydberg gate quantum processor to a large number of qubits. Limits are derived showing that the number of qubits that can be directly connected by entangling gates with errors at the 10^-3 level using long-range Rydberg interactions between sites in an optical lattice, without mechanical motion or swap chains, is about 500 in two dimensions and 7500 in three dimensions. A scaling factor of 60 at a smaller number of sites can be obtained using collective register encoding in the hyperfine ground states of the rare-earth atom holmium. We present a detailed analysis of operation of the 60-qubit register in holmium. Combining a lattice of multiqubit ensembles with collective encoding results in a feasible design for a 1000-qubit fully connected quantum processor.