Optimizing quantum sensing networks via genetic algorithms and deep learning

TL;DR

This work tackles the optimization of graph-structured quantum sensing networks for estimating weak magnetic fields by combining a genetic algorithm with a spectral-sensitivity objective $D_n$ and a quantum Fisher information benchmark. It shows that optimal topologies, identified via $D_n$, can yield large metrological gains at small to moderate sizes, but the quantum Fisher information exhibits non-monotonic, often diminishing returns with increasing system size, especially under Kac scaling that enforces extensive energy. The authors reveal that even-odd oscillations and phase-space interference underlie these quantum features, as demonstrated by Husimi $Q$-function analyses. To scale to larger graphs, they train deep neural networks on GA data to extrapolate $D_n$ and $F_Q$, enabling efficient performance estimation without full quantum simulations. Overall, the study demonstrates the value of topology-aware optimization and hybrid evolutionary-learning strategies for designing high-performance quantum sensing networks.

Abstract

We investigate the optimization of graph topologies for quantum sensing networks designed to estimate weak magnetic fields. The sensors are modeled as spin systems governed by a transverse-field Ising Hamiltonian in thermal equilibrium at low temperatures. Using a genetic algorithm (GA), we evolve network topologies to maximize a perturbative spectral sensitivity measure, which serves as the fitness function for the GA. For the best-performing graphs, we compute the corresponding quantum Fisher information (QFI) to assess the ultimate bounds on estimation precision. To enable efficient scaling, we use the GA-generated data to train a deep neural network, allowing extrapolation to larger graph sizes where direct computation becomes prohibitive. Our results show that while both the fitness function and QFI initially increase with system size, the QFI exhibits a clear non-monotonic behavior - saturating and eventually declining beyond a critical graph size. This reflects the loss of superlinear scaling of the QFI, as the narrowing of the energy gap signals a crossover to classical scaling of the QFI with system size. The effect is reminiscent of the microeconomic law of diminishing returns: beyond an optimal graph size, further increases yield reduced sensing performance. This saturation and decline in precision are particularly pronounced under Kac scaling, where both the QFI and spin squeezing plateau or degrade with increasing system size. We also attribute observed even-odd oscillations in the spectral sensitivity measure and QFI to quantum interference effects in spin phase space, as confirmed by our phase-space analysis. These findings highlight the critical role of optimizing interaction topology - rather than simply increasing network size - and demonstrate the potential of hybrid evolutionary and learning-based approaches for designing high-performance quantum sensors.

Figures (15)

• Figure 1: Genetic algorithm workflow.
• Figure 2: Example of graph evolution through genetic operations for $N = 4$ nodes. Left to right: (1) Parent 1 is a linear (path) graph with edges $(1,2), (2,3), (3,4)$. (2) Parent 2 is a square graph with edges $(1,2), (2,3), (3,4),(1,4)$. (3) Child (Crossover) combines common edges from both parents, and the Mutated Child further modifies the child by adding another random edge (1,4) (red colored), increasing the graph's connectivity.
• Figure 3: Optimal connected graph structures obtained via the genetic algorithm for $N = 1$ to $N = 12$. Each graph with the maximum $D_n$ value maximizes the QFI under the given model, reflecting the topologies most sensitive to magnetic field estimation. These graphs are obtained for the parameters $T=0.08$, $h=0.05$, $p=100$, and $n_G=15$.
• Figure 4: $D_n$ (blue) and QFI (red) as a function of number of vertices $N$ for the optimal graphs shown in Fig. \ref{['fig:graphs']} and using two different values of temperature (a) $T=0.08$ and (b) $T=2$ when the magnetic field is set to $h=0.05$. The rest of the parameters are fixed at $p=100$ and $n_G=15$.
• Figure 5: QFI as a function of the fitness function $D_n$ for the optimal graphs shown in Fig. \ref{['fig:graphs']}. The parameters are set to $T=0.08$, $h=0.05$, and $N=12$. The rest of the parameters are the same as in Fig. \ref{['fig:DifferentT']}.