Configuration-dependent precision in magnetometry and thermometry using multi-qubit quantum sensors

TL;DR

This work addresses configuring a four-qubit quantum sensor in the transverse-field Ising model to optimize magnetometry and thermometry. By computing the quantum Fisher information for six graph-based configurations under ferromagnetic and antiferromagnetic couplings, it identifies topology-dependent regimes: sparse graphs like $P_4$ optimize weak-field magnetic sensing, while highly connected graphs like $K_4$ excel under strong fields and in thermometry; in the antiferromagnetic regime, the pan graph favors magnetometry and $C_4$ favors thermometry. A spectral-deformation measure $D_n(h)$ is introduced as a simple heuristic to predict metrological sensitivity from low-energy spectrum changes. The results provide concrete design rules linking graph topology, energy spectrum structure, and degeneracy to sensing performance, with potential applications in scalable quantum sensing networks.

Abstract

We study the performance of quantum sensors composed of four qubits arranged in different geometries for magnetometry and thermometry. The qubits interact via the transverse-field Ising model with both ferromagnetic and antiferromagnetic couplings, maintained in thermal equilibrium with a heat bath under an external magnetic field. Using quantum Fisher information, we evaluate the metrological precision of these sensors. For ferromagnetic couplings, weakly connected graphs (e.g., the chain graph, P_4) perform optimally in estimating weak magnetic fields, whereas highly connected graphs (e.g., the complete graph, K_4) excel at strong fields. Conversely, K_4 achieves the highest sensitivity for temperature estimation in the weak-field regime. In the antiferromagnetic case, we uncover a fundamental trade-off dictated by spectral degeneracy: configurations with non-degenerate energy spectra - such as the pan-like graph (three qubits in a triangle with the fourth attached) - exhibit strong magnetic field sensitivity due to their pronounced response to perturbations. In contrast, symmetric structures like the square graph, featuring degenerate energy levels (particularly ground-state degeneracy), are better suited for precise thermometry. Notably, our four-qubit sensors achieve peak precision in the low-temperature, weak-field regime. Finally, we introduce a spectral sensitivity measure that quantifies energy spectrum deformations under small perturbations, providing a simple heuristic indicator of metrological sensitivity.

Figures (12)

• Figure 1: Model representation of our quantum sensing protocol. The quantum sensor, modeled as a network of four coupled spins, simultaneously exposed to a thermal bath at temperature $T$ and an external weak magnetic field with magnitude $\Vec{h}$, directed along $x$-axis. The sensor functions as a probe to estimate unknown parameters, including $T$ and $h$. The different configurations of the quantum sensor are given in Fig. \ref{['fig2']}.
• Figure 2: Graphical representation of the structural arrangement of four spins in six different configurations: (a) Chain ($P_4$), (b) $4$-cycle square ($C_4$), (c) $4$-cycle with a diagonal ($Sd_4$), (d) Complete ($K_4$), (e) Triangle with a pendant vortex (pan graph), (f) Star-tree graph ($S_3$). In each configuration, the spins are coupled to a thermal bath at temperature $T$ and subjected to an external magnetic field of strength $h$. The vertices, represented by solid purple spheres, denote the spin qubits, while the dashed lines indicate the interactions between them.
• Figure 3: Energy eigenvalue spectra for six different configurations of the quantum sensor in the presence of a weak external magnetic field ($h = 0.08$). The top panel corresponds to ferromagnetic coupling ($J = 1$) and the bottom panel to antiferromagnetic coupling ($J = -1$). The legends refer to six different configurations of the $4-$qubit quantum sensor, which are given in Fig. \ref{['fig2']}.
• Figure 4: The QFI ($F_h$) for magnetic field ($h$), is plotted as a function of the magnetic field strength $h$ for different quantum sensor configurations at temperatures $T = 0.04$ (a) and $T = 0.08$ (b). The different colors, such as red, green, blue, magenta, cyan, and orange, represent the configurations $P_4$, $C_4$, Pan, $Sd_4$, $K_4$, and $S_3$, respectively. All plots correspond to the ferromagnetic coupling ($J = 1$).
• Figure 5: The QFI ($F_T$) temperature estimation is plotted as a function of temperature $T$ for two values of magnetic fields $h = 0.45$ (a) and $h = 0.55$ (b). The different colors, such as red, green, blue, magenta, cyan, and orange, represent the configurations $P_4$, $C_4$, Pan, $Sd_4$, $K_4$, and $S_3$, respectively. All plots correspond to the ferromagnetic coupling ($J = 1$).