Quantum Amplitude Estimation for Wright-Fisher Parameter Inference: Theoretical Query Complexity Advantages Conditional on Efficient Oracle Construction

We establish theoretical foundations for applying quantum amplitude estimation to Wright-Fisher population genetics, demonstrating that quantum methods could provide substantial query complexity improvements-reducing sample requirements from [Formula: see text] to [Formula: see text]- conditional on efficient oracle construction. Through paired classical simulations comparing high-variance versus low-variance estimation strategies across extensive computational replicates, we isolate the effects of variance reduction on evolutionary dynamics while validating theoretical predictions with exceptional precision. Our comprehensive statistical analysis confirms distributional equivalence between methods through consensus across multiple independent tests, including Bayesian model comparison and generational Kolmogorov-Smirnov testing. Overlapping confidence intervals for fixation probability validate that query complexity reduction does not degrade evolutionary prediction quality-the central requirement for practical quantum advantage. Fixation timing distributions remain statistically indistinguishable, demonstrating that variance reduction preserves both stochastic establishment dynamics and deterministic sweep phases across the complete evolutionary trajectory. These theoretical advantages depend entirely on three unresolved prerequisites: (i) oracle construction overhead must remain polynomial in problem parameters, requiring either quantum random access memory (qRAM)-experimentally undemonstrated at practical scales-or problem-specific circuits for fitness landscape encoding; (ii) fault-tolerant quantum hardware with gate error rates orders of magnitude below current technology; (iii) scalable quantum architectures maintaining coherence across the required computational depth. Break-even analysis establishes that without efficient oracle construction, query complexity advantages may be offset by oracle overhead, resulting in marginal or zero net computational benefit. This work contributes: (a) rigorous characterization of variance propagation through evolutionary dynamics with near-perfect scaling law validation, (b) identification of weak-selection, high-noise regimes where quantum variance reduction provides maximum benefit, and (c) transparent delineation of necessary versus sufficient conditions for practical quantum advantage. Our framework establishes that reduced estimation variance, if achievable through quantum or other advanced methods, yields measurable evolutionary benefits while explicitly acknowledging that translating theoretical query advantages into end-to-end computational speedup requires coordinated breakthroughs in quantum algorithmic primitives and hardware capabilities.