Time complexity in preparing metrologically useful quantum states

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

Figures (2)

• Figure 1: Comparison of the time-scaling exponent $\beta$ from our theoretical bound ($t \gtrsim L^{\beta}$) with that of the fastest known protocols ($t \sim L^{\beta}$), ignoring sub-polynomial corrections. The task is to prepare a metrologically useful entangled state ($F_Q \sim N^{1+\gamma}$) using a Hamiltonian with $1/r^{\alpha}$ spin-spin interactions. We compare $\beta_{\text{bound}}$ (solid lines) and $\beta_{\text{protocol}}$ (dashed lines) for both $\gamma=1$ and $\gamma=0.5$. Up to sub-polynomial corrections, our bound is optimal for $\gamma=1$ (where the exponents match) with any $\alpha>0$ or for $0<\gamma<1$ with $\alpha > (2-\gamma)d$.
• Figure 2: Optimal spin-squeezing time $t_{\text{opt}}$ (left) and the corresponding QFI $F_{Q}^{\text{opt}}$ (right) as a function of the number of spins $N$ for three spin squeezing protocols: (a) two-axis twisting (TAT), (b) twist-and-turn (TnT), and (c) one-axis twisting (OAT). The data points are fitted using the following models and the fitted models are shown as solid lines: $\chi t_{\text{opt}} = A N^{-2/3}$ with $A = 1.144 \pm 0.003$ for OAT, $\chi t_{\text{opt}} = A \ln(N)/N$ for TAT and TnT, with $A = 0.4730 \pm 0.0009$ and $A = 0.554 \pm 0.001$ respectively; $F_{Q}^{\text{opt}} = A N^2$ with $A = 0.3627 \pm 0.0014$ for TAT, $F_{Q}^{\text{opt}} = A N^{3/2}$ with $A = 1.32 \pm 0.02$ for TnT, and $F_{Q}^{\text{opt}} = A N^{5/3}$ with $A = 1.152 \pm 0.009$ for OAT.