Quantum King-Ring Domination in Chess: A QAOA Approach

TL;DR

The paper presents QKRD, a chess-derived, NISQ-scale benchmark to evaluate QAOA on structured, semantically meaningful problems. It systematically tests constraint-preserving mixers, warm-start initializations, and risk-aware objectives across 5,000 instances, highlighting substantial gains from problem-informed design. Key findings show constraint-preserving mixers remove penalty tuning and accelerate convergence, warm-starts yield large energy improvements, CVaR is not advantageous for this structure, and QAOA surpasses greedy and random baselines in coverage. By providing open-source data and artifacts, the work underscores the importance of structured benchmarks for meaningful, reproducible NISQ research.

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is extensively benchmarked on synthetic random instances such as MaxCut, TSP, and SAT problems, but these lack semantic structure and human interpretability, offering limited insight into performance on real-world problems with meaningful constraints. We introduce Quantum King-Ring Domination (QKRD), a NISQ-scale benchmark derived from chess tactical positions that provides 5,000 structured instances with one-hot constraints, spatial locality, and 10--40 qubit scale. The benchmark pairs human-interpretable coverage metrics with intrinsic validation against classical heuristics, enabling algorithmic conclusions without external oracles. Using QKRD, we systematically evaluate QAOA design choices and find that constraint-preserving mixers (XY, domain-wall) converge approximately 13 steps faster than standard mixers (p<10^{-7}, d\approx0.5) while eliminating penalty tuning, warm-start strategies reduce convergence by 45 steps (p<10^{-127}, d=3.35) with energy improvements exceeding d=8, and Conditional Value-at-Risk (CVaR) optimization yields an informative negative result with worse energy (p<10^{-40}, d=1.21) and no coverage benefit. Intrinsic validation shows QAOA outperforms greedy heuristics by 12.6\% and random selection by 80.1\%. Our results demonstrate that structured benchmarks reveal advantages of problem-informed QAOA techniques obscured in random instances. We release all code, data, and experimental artifacts for reproducible NISQ algorithm research.

Figures (5)

• Figure 1: QKRD optimization example comparing a strong queen move and a much weaker pawn push against a black king on $d4$. The inner ring $R_1$ (blue, 8 squares adjacent to the king) and outer ring $R_2$ (red, Chebyshev distance 2) define the tactical target zones. Green circles mark ring squares attacked by the moving piece from its destination square for each candidate move. The weighted objective $\alpha_1 \cdot |R_1 \cap \text{attacks}| + \alpha_2 \cdot |R_2 \cap \text{attacks}|$ with $\alpha_1 > \alpha_2$ strongly favors the queen move Qd2+ over the pawn push f4, making the effect of king-ring dominance visually clear.
• Figure 2: Mixer comparison showing convergence speed across different mixer types.
• Figure 3: Warm-start ablation showing convergence and energy distributions.
• Figure 4: CVaR vs expectation energy distributions on lifted temporal instances.
• Figure 5: Coverage distributions for QAOA, greedy, and random across 5000 positions.