Time complexity in preparing metrologically useful quantum states

We investigate the fundamental time complexity, as constrained by Lieb-Robinson bounds, for preparing entangled states useful in quantum metrology. We relate the minimum time to the Quantum Fisher Information ($F_Q$) for a system of $N$ quantum spins on a $d$-dimensional lattice with $1/r^\alpha$ interactions with $r$ being the distance between two interacting spins. We focus on states with $F_Q \sim N^{1+\gamma}$ where $\gamma \in (0,1]$, i.e., scaling from the standard quantum limit to the Heisenberg limit. For short-range interactions ($\alpha > 2d+1$), we prove the minimum time $t$ scales as $t \gtrsim L^\gamma$, where $L \sim N^{1/d}$. For long-range interactions, we find a hierarchy of possible speedups: $t \gtrsim L^{\gamma(\alpha-2d)}$ for $2d < \alpha < 2d+1$, $t \gtrsim \log L$ for $(2-\gamma)d < \alpha < 2d$, and $t$ may even vanish algebraically in $1/L$ for $\alpha < (2-\gamma)d$. These bounds extend to the minimum circuit depth required for state preparation, assuming two-qubit gate speeds scale as $1/r^\alpha$. We further show that these bounds are saturable, up to sub-polynomial corrections, for all $\alpha$ at the Heisenberg limit ($\gamma=1$) and for $\alpha > (2-\gamma)d$ when $\gamma<1$. Our results establish a benchmark for the time-optimality of protocols that prepare metrologically useful quantum states.