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A local Hamiltonian is built from interactions with restricted support. In quantum-information terminology, a k-local Hamiltonian has terms acting on at most a fixed number $k$ of subsystems. Geometric locality adds that those subsystems are nearby in a specified spatial arrangement.

A nearest-neighbor spin chain satisfies both requirements. A sum of interactions between arbitrary distant pairs is two-local in the support-counting sense but need not be geometrically local. This distinction matters when estimating propagation, simulation cost and hardware connectivity. Locality constrains individual terms, not whether the evolved state stays unentangled: repeated local interactions can spread correlations across the system. The geometry, interaction range and strength assumptions must accompany a dynamical locality claim.

Each summand of $J_X$ acts on one qubit, while each summand $Z_i{Z}_j$ of $J_Z$ acts on two. Therefore $J=J_X+J_Z$ is a 2-local Hamiltonian. We haveSplit $t$ into $r$ steps of length $\delta =t/r$ and use the second-order product formulaFor one step, $\Lambda =O(\delta n^2)$ in the stated estimate, so the spectral-norm error is $O({\delta }^3{n}^6)$. The error bound for a product of unitary operators makes the total errorIt is therefore enough to choosewith $r$ also large enough that the small-step estimate applies.

All $Z_i{Z}_j$ terms commute. A factor $e^{-i\delta Z_i{Z}_j}$ uses a constant-size circuit of two controlled-NOT gates and one phase gate, up to a global phase, so one product-formula step costs $O(n^2)$ gates. The complete Hamiltonian simulation consequently has size