Time complexity in preparing metrologically useful quantum states

TL;DR

This paper establishes fundamental bounds on the minimum time required to prepare entangled quantum states useful for metrology, considering interaction range and quantum Fisher information scaling.

Contribution

It derives time complexity bounds for preparing quantum metrological states based on Lieb-Robinson constraints, extending to long-range interactions and circuit depth.

Findings

Minimum preparation time scales with system size and interaction range.

Hierarchy of speedups depending on interaction decay exponent.

Bounds are achievable for certain interaction regimes and Fisher information scalings.

Abstract

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.