Many-body $k$-local ground states as probes for unitary quantum metrology

TL;DR

This work addresses the challenge of achieving Heisenberg-limited metrology with experimentally accessible interactions. By studying ground states of permutation-invariant, $k$-local Hamiltonians and leveraging quantum Fisher information, it shows that typical random ground states exhibit Heisenberg scaling even when only few-body terms are used. A key advance is a dynamic programming framework that efficiently computes the QFI in the symmetric (Dicke) subspace, enabling analysis at large $N$. The authors also reveal a quantitative tradeoff between the Hamiltonian gap, which quantifies preparation hardness, and the QFI, and they establish conditions under which Haar-random-like metrological advantages can be approached with physically plausible few-body interactions.

Abstract

Multipartite quantum states saturating the Heisenberg limit of sensitivity typically require full-body correlators to be prepared. On the other hand, experimentally practical Hamiltonians often involve few-body correlators only. Here, we study the metrological performances under this constraint, using tools derived from the quantum Fisher information. Our work applies to any encoding generator, also including a dependence on the parameter. We find that typical random symmetric ground states of $k$-body permutation-invariant Hamiltonians exhibit Heisenberg scaling. Finally, we establish a tradeoff between the Hamiltonian's gap, which quantifies preparation hardness, and the quantum Fisher information of the corresponding ground state.

Figures (6)

• Figure 1: Average and fluctuations of the QFI as a function of $N$ (even) over 2000 ground states of $\mathrm{H}$, constructed with randomly sampled coefficients $\Gamma _{abc}$ drawn from a normal distribution $\mathcal{N}(0,1)$, for various $k$-body interactions. Solid curves denote sample averages; shaded regions indicate $\pm$ standard error of the mean (SEM). The red dashed line shows the reference value $N(N+1)/3$. Inset: Frequency histograms of the QFI at $N=40$ for various $k$-body interactions. Vertical dashed lines mark the corresponding sample averages; light spans denote $\pm$SEM.
• Figure 2: Energy gap and corresponding QFI for $10^5$ ground states of random two-body Hamiltonians for $N=10$ particles and $k=2$. Eq. \ref{['H_sys']} (blue dots). Orange dots correspond to Hamiltonians with only nonzero coefficients, $\Gamma_{200},\Gamma_{020}, \Gamma_{002}$. The star corresponds to Eq. \ref{['eq:H_squeezing']}, which displays a gap of $12/(N(N+2))$. Plots for higher $k$ can be found at the End Matter.
• Figure 3: The QFI as a function of $N$ for the encoding generator $K_\theta = \cos(\theta)\sigma _{z}+\sin(\theta)\sigma _{x}$ (cf. Fig. \ref{['H_X-random_sampling']}) for random $\theta$. The probe states are the ground states of $H$, constructed with randomly sampled coefficients $\Gamma _{abc}$ drawn from a normal distribution $\mathcal{N}(0,1)$, for various $k$-body interactions. For each value of $k$, $2000$ random instances were sampled.
• Figure 4: Energy gap and corresponding QFI for $10^5$ ground states of random two-body Hamiltonians for $N=10$ particles and different order of interactions $k$ (blue dots). Orange dots correspond to Hamiltonians with only nonzero coefficients, $\Gamma_{k00},\Gamma_{0k0}, \Gamma_{00k}$.
• Figure 5: The QFI as a function of $N$ for the case where the encoding Hamiltonian is $G_\theta = \cos(\theta)\sigma _{z}+\sin(\theta)\sigma _{x}$. The probe states are the ground states of system's Hamiltonian, constructed with randomly sampled coefficients drawn from a normal distribution $\mathcal{N}(0,1)$, for various $k$-body interaction terms. For each value of $k$, $2000$ random instances were sampled.

Theorems & Definitions (2)

• Theorem 1
• Lemma 1