An End-to-End Hybrid Quantum--Classical Sampling Workflow for Discrete Markov Random Fields: A Reproducible Case Study

Sampling from discrete Markov random fields (MRFs) is a hard problem. We study amplitude-encoded i.i.d. sampling for small MRFs where 2 n target probabilities are precomputed classically. This removes quantum exponential speedup but allows a clean comparison against classical MCMC based on independent circuit samples (τ about 1). Across 60 instances spanning five graph families (1k-step burn-in, 3k retained samples), the mean ESS ratios of Quantum to Single-Site Gibbs, Block Gibbs, Tuned-Block, and Parallel Tempering are 16.35, 7.29, 1.82, and 1.79, showing modern classical samplers substantially close this gap. Amortizing O(2 n) preprocessing into wall-clock time, exact inverse-CDF sampling yields 17.7 M ESS/s versus 488 K ESS/s for the quantum sampler (36 times mean rate, 153 times per-instance), confirming no wall-clock advantage. We characterize MCMC autocorrelation costs and benchmark amplitude-encoded state preparation at n in 8,10,12. An MPS scaling study (n 40) shows bond dimension χ=32 achieves F=0.721 plus or minus0.059 at n=40. Finally, a matched-budget VQC vs. MPS comparison at n in 8,10,12 shows VQC fidelities fall far below MPS: (F VQC, F MPS) = (0.31, 0.99), (0.21, 0.96), (0.17, 0.88) at compressions 10.7 times, 34.1 times, and 113.8 times.