Star Topology Optimizes the Charging Power of Quantum Batteries

Abstract

Quantum batteries are quantum systems that store energy and deliver it on demand, and their practical value hinges on how fast they can be charged. While collective charging protocols and global control are known to enhance charging power, it remains unclear how the battery's internal interaction architecture itself constrains performance. Here we study interacting fermionic batteries whose internal couplings are encoded by a graph adjacency matrix, charged via a simple interaction with an external fermionic device. We prove that the star topology maximises the early time charging power, which proxies the maximal average power - a widely used quantum battery quality metric. We substantiate the result numerically by an exhaustive sweep over all graphs with $N\leq 7$ vertices and by benchmarks against random graph ensembles at larger $N$. Our findings shed light on architecture as a controllable knob for fast charging and motivate hub-and-spoke designs in scalable quantum-battery platforms.

Figures (11)

• Figure 1: Spectral envelope ratios for random graph families show no exceedances of the star benchmark. Mean values and standard deviations of $\max_{\varepsilon_k<0}(\mathbf{1}^{\top} u_k)^2 / w_-^{(S)}$ across graph sizes $N \in \{10, 15, 20, 30, 40, 50, 60\}$ for six random graph models (500 samples per model per size). The red dashed line marks the star benchmark (ratio = 1). All tested architectures remain systematically below the benchmark, with ratios decreasing as system size increases, providing strong empirical evidence that no random topology surpasses the star's early-time charging performance.
• Figure 2: Distribution analysis confirms systematic underperformance relative to the star benchmark. Histograms of spectral envelope ratios $\max_{\varepsilon_k<0}(\mathbf{1}^{\top} u_k)^2 / w_-^{(S)}$ at the largest tested size ($N=60$) for six random graph families (500 samples each). Each panel shows the ratio distribution for: (top row) Erdős-Rényi graphs with edge probabilities $p=0.10$ and $p=0.20$; (middle row) uniform random trees and Barabási-Albert preferential attachment with $m=2$; (bottom row) unbalanced stochastic block model and Barabási-Albert with $m=3$. The red vertical line marks the star benchmark (ratio = 1). All distributions peak well below unity and exhibit minimal tail overlap with the benchmark, demonstrating that the star architecture's superiority is robust across diverse random topologies. Statistical summaries (max and mean values) are displayed for each family.
• Figure 3: $P_{\max}$ across graph topologies and system sizes. Maximum charging power $P_{\max}$ as a function of the number of sites $N$ for all connected interaction graphs, with $N=3,\dots,7$. Each blue dot corresponds to a distinct graph topology. The star graph ($S_N$), the perturbed star graph (second-best for $N=7$; see Fig. \ref{['fig:top7-star-perturbed']}), the path graph ($P_N$) and the complete graph ($K_N$) are highlighted. The two panels correspond to different values of $h$: $h = 0$ (top) and $h = 1$ (bottom), with $\omega = \kappa = 1$.
• Figure 4: $P_{\max}$ and $w_\mathrm{min}$ as a function of the topological index at fixed system size.$N=7$ for all connected interaction graphs. Canonical graph topologies are highlighted: star ($S_N$), perturbed star, path ($P_N$), and complete graph ($K_N$). The index $G$ follows the canonical ordering of the NetworkX graph atlas. The first plot corresponds to $h=0$, the second to $h=1$. The plot of $w_\mathrm{min}$ is for $h=0$.
• Figure 5: Top three interaction topologies for $N = 7$.

Theorems & Definitions (31)

• Claim 1
• Lemma 1
• proof
• Theorem 1: Star optimality at $h=0$
• proof
• Remark 1
• Conjecture 1: Star optimality for the uniform overlap
• Definition 1: Regular graphs and spectral expanders
• Proposition 1: Regular graphs have zero uniform overlap in nontrivial modes
• proof