Quantum batteries with K-regular graph generators: A no-go for quantum advantage

TL;DR

This work analyzes quantum batteries built from stabilizer Hamiltonians tied to $K$-regular graphs to assess potential quantum advantages. It shows a no-go result: for a local battery initialized along a fixed spin direction and charged via a $K$-regular graph Hamiltonian with $K\ge 2$, the extractable work scales linearly with system size $N$, ruling out superlinear quantum advantage. The authors also study how regularity $K$, partial accessibility of the battery, and collective charging with power-law couplings across multiple $K$ affect performance, finding no superlinear scaling of average power. Collectively, these results clarify fundamental limits of graph-structured quantum batteries and delineate how graph connectivity and subsystem accessibility govern thermodynamic performance. The findings impose precise bounds on when quantum advantages can be expected in graph-based energy storage paradigms.

Abstract

Regular graphs find broad applications ranging from quantum communication to quantum computation. Motivated by this, we investigate the design of a quantum battery based on a K-regular graph, where K denotes the number of edges incident on each vertex. We prove that a 0-regular graph battery exhibits extractable work that scales linearly with the system-size when charged using a K-regular graph. This linear scaling is shown to persist even when the charging is implemented via a collective K-regular charger with power-law decaying interactions. While no superlinear scaling is observed, the work output is found to improve systematically with increasing regularity K. Furthermore, by introducing the notion of the fraction of extractable work when only subsystems are accessible, we identify this fraction to be independent of system-size if the battery is prepared in the down-polarized product state. This independence breaks down when the battery is oriented along the x- and y-directions of the Bloch sphere.

Figures (6)

• Figure 1: (Color online.) In a $K$-regular ($K$ even) connected graph $G_{(N,K)}$ of $N$ nodes, each node, indexed by $i$ ($i=1,2,\cdots,N$), is connected to $K$ nodes such that the links $\{(i,i+j),(i,i-j);j=1,2,\cdots,K/2\}\in G_{(N,K)}$. Examples with (a) $K=2$, (b) $K=4$, and (c) $K=6$ are shown for arbitrary $N$. Note that each $G_{(N,K)}$ corresponds to a stabilizer Hamiltonian $H_{(N,K)}$ (see Eq. (\ref{['eq:HNK']})), while all three graphs shown in (a)-(c) contribute to constitute $H_C=\sum_{K=2}^6J_K H_{(N,K)}$, with $6\leq K_{\max}=N-2$$(N-1)$ for even (odd) $N$.
• Figure 2: (Color online.) Variations of work stored over the system-size, $N^{-1}W^A_{(N,2)}$ (vertical axis) with time, $t$ (horizontal axis) for the charger, $H_{(N,2)}$. Solid, dashed and dashed-dot lines signify the orientation of the initial battery Hamiltonian, $X,Y$, and $Z$ respectively. Note that the time period of $N^{-1}W^A_{(N,2)}$ is $t=\pi$ for $A=Y$ and $Z$ with the maximum value $2$ occur at $t=\pi/2$ whereas $(N^{-1}W^X_{(N,2)})_{\max}=1$ at $t=\pi/4$. All the axes are dimensionless.
• Figure 3: (Color online.) $N^{-1}W^X_{(N,K)}$ (abscissa) is plotted against $t$ (ordinate) for different $K$-regular charging Hamiltonian with (a) $N=4$, (b) $N=6$, (c) $N=8$, and (d) $N=10$. For a particular $N$, different line styles (solid, dashed, dashed-dotted and dotted) indicate different $K$ values. Note that for $K=K_{\max}$, $N^{-1}W^X_{(N,K)}$ shows different behavior than $K<K_{\max}$. At $t=\frac{\pi}{4}$, for $N=4$ and $8$ the maximum of $N^{-1}W^X_{(N,K)}$ is $2$ while for $N=6$ and $10$, it is $1$. All the axes are dimensionless.
• Figure 4: (Color online.) (a) Average work $\overline{W}^X_{(N,K)}$ (y-axis) with respect to regularity, $K$ (x-axis). We find that $\overline{W}^X_{(N,K)}$ increases with regularity, $K$ and saturates to a finite value, depicting the role of $K$ in storing average work within a time domain. (b) Average power, $\overline{P}^A_{(N,K)}$ (y-axis) against $K$ (x-axis) for the battery Hamiltonian $H_B^X$ and $H_B^Y$ with squares and stars respectively. All the axes are dimensionless.
• Figure 5: (Color online.) Fraction of the extractable energy, $\overline{R}^A_m$ (vertical axis) with the ratio, $m/N$ (horizontal axis) for the regularity $K=2$. The subplots are for the battery Hamiltonian with (a) $A=X$, (b) $A=Y$, and (c) $A=Z$. For a particular battery Hamiltonian $H_B^A$, different line-points indicates different $N$ ranging from $N=6 \, (\text{circles}),8\, (\text{squares}),10$ (left-arrowed triangles) and $12$ (right-arrowed triangles). Note that $\overline{R}^Z_m$ is independent of $N$ differ from $A=X$ and $Y$ where there is a dependency of $N$. All the axes are dimensionless.

Theorems & Definitions (3)

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