Quantum-Assisted Graph Domination Games

TL;DR

The paper investigates quantum advantage in a 1-step graph-domination game on cycle graphs using theory, numerical optimization, and NISQ experiments. By sharing Bell-state entanglement and applying site-dependent local rotations, the authors derive explicit quantum strategies that reproduce known bounds for small graphs and generalize to larger cycles, with an analytically tractable expression for the domination performance $D_n(\theta)$. They identify optimal angle schedules, show how the increment $\theta_n$ transitions between regimes, and validate these predictions through simulations on classical computers and real quantum hardware. The results demonstrate that current NISQ devices can realize a meaningful quantum advantage in this nonlocal coordination task, albeit with hardware-imposed limitations, highlighting the potential for practical quantum-assisted coordination in networked settings and motivating extensions to broader graph classes and higher-dimensional entanglement.

Abstract

We study quantum advantage in the 1-step graph domination game on cycle graphs numerically, analytically and through the use of Noisy intermediate scale quantum (NISQ) processors. We find explicit strategies that realise the recently found upper bounds for small graphs and generalise them to larger cycles. We demonstrate that NISQ computers realise the predicted quantum advantages with high accuracy.

Figures (5)

• Figure 1: Classical and quantum-assisted approaches to the graph domination game. (\ref{['ClassPlay']}) shows a play-through of the classical domination game. The players Alice (A) and Bob (B) are randomly placed (on sites 1 and 5, in this example). They then use a pre-agreed strategy to decide their moves. In our example, A moves from 1 to 2 and B moves from 5 to 4, which corresponds to the optimal classical strategy shown in Fig. 4 of Ref. PiotrBasePaper. They then check how much of the graph is dominated between them (shaded area). In this case, they dominate all 5 nodes of the graph. (\ref{['2ChcoiceCirc']}) shows the quantum circuit A and B can use to gain quantum advantage. Before A and B get separated (left of the dashed line) they perform a joint operation on their two qubits that places them in the entangled state of Eq. (\ref{['eq:Bell']}). Afterwards the players get assigned their starting nodes and A(B) rotates her qubit around the $y$ axis by an angle $\theta_i^A$ ($\theta_j^B$) that depends on the index $i$ ($j$) of the site she is in. She then measures her qubit in the computational basis and uses the result to decide whether to move clockwise (1) or anti-clockwise (0).
• Figure 2: Optimization of the angle increment $\theta$ for maximising the domination number $D(n,\theta)$. (a) Domination number as a function of the angle for $n=9,10,11$, showing a shift in the optimal value of $\theta$ from the one given in Eq. (\ref{['AngSepEq']}) ($n=9$) to the one given in Eq. (\ref{['AngSepEq2']}) ($n=11$). (b) Optimal value of $\theta$ as a function of $n$.
• Figure 3: Theoretical prediction of the average domination number achieved by our optimised quantum strategy for cycle graphs compared to a coin-tossing strategy (random choice) and to the optimal classical strategy, when known. In each case, the domination number is given as a function of the number $n$ of vertices in the graph.
• Figure 4: Performance of optimal strategies for the graph domination game on a 5-site cycle. The dashed lines indicate the predicted averages over many runs for the optimal classical and quantum strategies, as indicated. The solid lines show: simulations of the optimal classical and quantum strategies using a classical computer; and a simulation of the optimal quantum strategy using the best found superconducting quantum processor, IBM Kyiv.
• Figure 5: Performance of optimal strategies for the graph domination game on 5, 6 and 7 cycle graphs simulated on different quantum processors. The dashed line indicates the predicted average for the optimal classical strategy and the solid line indicates the optimal quantum strategy. The averaging is over $\sim 1000000$ runs in each case. The smaller lines represent error bars, estimated by the standard deviation of each averaged data set (a single line is visible due to the extremely small standard deviation, $< 10^{-3}$ in all cases; the inset shows the zoomed-in error bar for two of the quantum processors).