Quantum King-Ring Domination (QKRD)

• Quantum King-Ring Domination (QKRD) is a structured, chess-inspired benchmark for evaluating QAOA on constrained quantum optimization problems.
• It leverages a detailed combinatorial design with one-hot and gating constraints, mapping real chess positions to QUBO formulations using up to 40 qubits.
• Benchmark experiments demonstrate that constraint-preserving mixers and warm-start strategies markedly improve convergence speed and solution feasibility.

Quantum King-Ring Domination (QKRD) is a structured NISQ-scale quantum optimization benchmark derived from chess, designed for comprehensive evaluation of the Quantum Approximate Optimization Algorithm (QAOA) on instances exhibiting semantic constraints and spatial locality. QKRD addresses the limitations of synthetic benchmarks by leveraging rich, human-interpretable chess tactical positions and by providing intrinsic validation metrics without reliance on classical oracles. The benchmark encompasses 5,000 real-game instances with binary decision variables, one-hot and gating constraints, and QUBO encodings up to 40 qubits, enabling rigorous algorithmic comparisons and reproducibility in quantum algorithm research (Stenzel et al., 1 Jan 2026).

1. Formal Definition and Combinatorial Structure

QKRD instances are centered on a local Region of Interest (ROI), typically a $5 \times 5$ window, around the opponent’s king in a chess position. The core combinatorial structure consists of $K$ candidate primary moves—selected as strictly increasing the so-called king-ring coverage—where $K \in \{6, 8, 12, 16\}$. In the “lifted temporal” variant, each primary move $m$ is further associated with $F$ follow-up moves, giving a total variable count $N = K + K \cdot F$. The variable set comprises

$$x_m \in \{0,1\},\quad m=1, ..., K;\qquad y_{m,f} \in \{0,1\},\quad f=1, ..., F,$$

with $x_m$ and $y_{m,f}$ encoding, respectively, the selection of primary and conditional follow-up moves.

The constraint set is defined by:

• One-hot constraint over primaries:

$\sum_{m=1}^K x_m = 1,$

• Gating constraints for each primary $m$:

$\sum_{f=1}^F y_{m,f} = x_m.$

The objective function leverages concentric “king-rings”:

• $R_1$: up to 8 squares at Chebyshev distance 1,
• $R_2$: squares at Chebyshev distance 2.

Let $c_{m}^{(1)}, c_{m}^{(2)}$ and $d_{m,f}^{(1)}, d_{m,f}^{(2)}$ count attacked squares in $R_1$ and $R_2$ respectively. A risk penalty $\rho_m$ (and $\rho_{m,f}$) penalizes attacking defended squares. The QUBO cost, which must be minimized, reads: $$C({\mathbf{x}}, {\mathbf{y}}) = -\alpha_1 \sum_{m} (c_m^{(1)} x_m + \sum_{f} d_{m,f}^{(1)} y_{m,f}) -\alpha_2 \sum_{m} (c_m^{(2)} x_m + \sum_{f} d_{m,f}^{(2)} y_{m,f}) +\beta \sum_{m} (\rho_m x_m + \sum_{f} \rho_{m,f} y_{m,f}) +\lambda_{\text{onehot}} \left(\sum_{m} x_m - 1\right)^2 +\lambda_{\text{gate}} \sum_{m} \left(\sum_{f} y_{m,f} - x_m\right)^2.$$

This is equivalently mapped to a quadratic Hamiltonian in the Pauli–$Z$ basis on $n = K + K\cdot F$ qubits,

$H_C = \sum_i h_i Z_i + \sum_{i<j} h_{ij} Z_i Z_j.$

2. QAOA Formulation and Ansatz Engineering

QKRD is implemented within the QAOA framework, encoding the QUBO cost function as the problem (cost) Hamiltonian: $$H_C = \sum_{\mathbf{z}} C(\mathbf{z})\,|\mathbf{z}\rangle \langle \mathbf{z}|,$$ over computational basis states $\mathbf{z} \in \{0,1\}^{n}$.

Three mixer strategies are evaluated:

1. Standard X-mixer:

$H_M^{(X)} = \sum_{i=1}^{n} X_i.$

This mixer explores the full Hilbert space and requires substantial penalty parameters $\lambda_{\text{onehot}}$, $\lambda_{\text{gate}}$ for constraint enforcement.

2. XY ring mixer: Partitions the qubits into disjoint one-hot blocks $B_\ell$,

$$H_M^{(\rm XY)} = \sum_{\ell=1}^B \sum_{i < j \in B_\ell} (X_i X_j + Y_i Y_j),$$

preserving Hamming weight and thus feasibility within each block.

3. Domain-wall mixer: Employs a domain-wall encoding, using a nearest-neighbour XY chain over each one-hot block:

$$H_M^{(\rm DW)} = \sum_{\ell=1}^{B} \sum_{i=1}^{|B_\ell|-1} (X_i X_{i+1} + Y_i Y_{i+1}).$$

Each QAOA step alternates between unitaries: $$U_M(\beta) = e^{-i\beta H_M},\quad U_C(\gamma) = e^{-i\gamma H_C}.$$

The QAOA ansatz is constructed at circuit depth $p=2$ in all experiments. Initial states analyzed include:

• Uniform superposition,
• Basis warm-start (computational basis encoding the greedy classical solution),
• Local superposition (modifying the greedy solution with small-angle XY rotations within blocks).

Classical optimization is performed via Adam for expectation objectives and COBYLA for Conditional Value-at-Risk (CVaR) objectives. For the latter, the target is

$$\mathrm{CVaR}_\alpha[H_C] = \mathbb{E}[H_C | H_C \leq \operatorname{VaR}_\alpha(H_C)],$$

estimated with $N_{\rm shots}=1024$ circuit measurements.

3. Benchmark Suite Construction and Evaluation Metrics

The QKRD suite contains 5,000 positions sampled from Lichess middlegame and endgame scenarios ensuring at least eight coverage-increasing moves in each $5\times5$ ROI. Qubit usage ranges from 6–18 (single-ply) to up to 40 (lifted temporal, extended windows).

Performance metrics include:

• Final QUBO energy $E^* = \min_{\boldsymbol\theta}\langle H_C \rangle$,
• Coverage: $$\alpha_1 |R_1 \cap \text{attacks}| + \alpha_2 |R_2 \cap \text{attacks}| - \beta \cdot \rho$$,
• Feasibility rate (fraction of outputs satisfying all constraints),
• Convergence speed (number of optimizer steps to 95% of final energy),
• Intrinsic validation versus:
  • Deterministic greedy heuristics,
  • Uniform random selection.

All code, data, QUBO instance generators, and experiment logs are publicly released for full reproducibility at https://github.com/lmu-cs/qkrd.

4. Comparative Algorithmic Analysis

Extensive experimentation on 3,000 single-ply and 200 lifted-temporal instances supports robust conclusions regarding QAOA variants and heuristic baselines:

• Constraint-preserving mixers (XY, domain-wall):
  • Achieve $\approx13$ steps faster convergence relative to X-mixer with optimal penalty weighting $(p < 2.5\times10^{-7},\ d \approx 0.5)$.
  • Guarantee feasibility in 100% of samples, versus 99% for X-mixer at penalty $\lambda=10$.
  • Final coverages are statistically indistinguishable across all mixers $(p > 0.3,\ |d| < 0.003)$.
• Warm-start strategies:
  • Basis warm-start reduces convergence by 45 steps compared to uniform initialization $(p \approx 7 \times 10^{-127},\ d = 3.35)$; local superposition yields 32-step improvement $(p \approx 6 \times 10^{-83},\ d = 2.15)$.
  • Final QUBO energy is markedly improved $(d > 8)$.
• CVaR optimization:
  • CVaR$\alpha$ minimization $(\alpha=0.05)$ leads to significantly worse energy $$(\Delta E = +16.1,\ p\approx8\times10^{-45},\ d=1.21)$$ and no measurable coverage gain.
  • As $\alpha \to 0.5$, performance converges to expectation-driven optimization, with coverage distributions unchanged.
• Intrinsic validation:
  • Across 5,000 positions (XY-mixer, basis warm-start): QAOA achieves mean coverage 5.32 squares; greedy 4.73 $(+12.6\%,\ p < 10^{-10})$; random 2.95 $(+80.1\%)$.
  • Coverage distributions are consistently right-shifted under QAOA relative to classical baselines.

5. Significance, Limitations, and Availability

The QKRD benchmark demonstrates that structured, semantically-informed quantum optimization benchmarks can reveal substantive advantages of problem-tailored QAOA algorithms unobservable in synthetic random instances. Specifically, constraint-preserving mixers (XY, domain-wall) not only eliminate the requirement for penalty term tuning but also substantially accelerate convergence with no loss in final solution quality. Warm-start strategies are particularly powerful in structured domains, cutting optimization steps by over 70%.

A crucial finding is that CVaR-QAOA, though theoretically motivated for improving rare-event solutions, can in fact degrade performance on QKRD—a result of practical and methodological interest.

By providing a reproducible, semantically rich test suite tailored for mid-scale NISQ quantum devices and systematically benchmarking classical versus quantum-influenced strategies, QKRD constitutes a critical resource for advancing algorithmic techniques in quantum optimization, with all necessary artifacts available for benchmarking and replication (Stenzel et al., 1 Jan 2026).