the $^3$P$_2$ manifold that can be coupled to both qubit states. For $^{87}$Sr, such addressing for single-qubit operations would thus mean either using an auxiliary lattice to trap states from the $^3$P$_2$ manifold, or using the upper dressed states for qubit storage.

Two-qubit gates can, in principle, be performed similarly to exisiting schemes for alkali atoms, making use of the spin-dependent potentials. In particular, exisiting schemes for controlled collisions can be used to produce controlled-phase gates for atoms in neighbouring sites [@spinlattice1]. This has been implemented experimentally in a proof-of-principle experiment with alkali atoms [@spinlattice2], but here we could take advantage of the 2D spin-dependent lattices without having to tune trapping lasers between fine-structure states.

These schemes can be seen to implement controlled-phase gates in three steps:

1.  The spin-dependent lattices for each state are shifted relative to each other so that atoms at a chosen distance, e.g., in neighbouring lattice sites, will come together on the same site if and only if they were originally in a specific combination of qubit states. For example, if we write the state of a pair of neighbouring qubits as ${|q_1 q_2 \rangle}$, where $q_1$ is the state of the first qubit and 