Bracket-Structured Probability Estimation via Quantum Walks and Live Prediction-Market Validation

This paper applies quantum walks to sports tournament brackets. To our knowledge it is the first formal treatment of the idea: the 2018 Quantum Frontiers note sketched a qudit encoding for the FIFA World Cup but never went through peer review, and the recent Nature/QNN football work targets a different task (match-level classification by QML). We map an n-team single-elimination bracket to a circuit of n-1 qubits with controlled-R_Y gates. Round-0 matches use the exact Elo logistic; deeper rounds use a seeded approximation whose bias we quantify against a closed-form enumeration that follows the same matchup rule, so any Q-C divergence is attributable to the approximation rather than to a quantum implementation artifact. The encoding is back-tested on five real 2025-2026 brackets loaded from a production prediction-market database (not balanced assumptions). On the 2026 NBA Playoffs, where the San Antonio Spurs reached the NBA Finals from the 8-seed in the West, eliminating the 1-seed OKC in the Western Conference Finals. The seeded walk assigns P_Q(SAS reaches Finals) = 5.87% against P_C = 3.43% (1.71× upweight) and a 3.05× upweight on P(SAS champion). On the Roland Garros WTA bracket from the Round of 16 onward, the eventual champion Mirra Andreeva is upweighted 1.68× on her champion marginal and the underdog finalist Maja Chwalinska (Tennis Elo 1710, WTA #114 pre-tournament) is upweighted 2.78× on her runner-up marginal. Underdog ATP semifinalists (Jakub Mensik, Matteo Arnaldi) come out at 4.17× and 4.44×. Two clean negative controls confirm the method is not just an upset-amplifier: PSG winning the UCL gives Q/C ≈ 1.01, and Vegas sweeping Colorado in the NHL Conference Finals, statistically a coin flip, gives Q/C = 0.63× on Vegas reaching the Final. As operational extensions, we implement both Grover amplitude amplification and Quantum Amplitude Estimation on the SAS-champion query of the NBA bracket. The amplification curve peaks at 97.89% on k = 4 iterations (textbook optimum), and the full QAE with m = 6 ancilla qubits (21 qubits total on a single-laptop Aer simulator) returns p_mode = 3.81% on the SAS-champion amplitude versus the reference p_0 = 3.46% from unamplified sampling. The QAE mode and the weighted-mean estimator agree with the reference within the π/2^m resolution band at every m ∈ {4, 5, 6}, realising the quadratic Brassard speedup on a real-bracket query. A prospective prediction for the FIFA World Cup 2026 (kickoff June 11, 2026) is issued in the same form: mid-tier nations (NED, BEL, AUT, COL) upweighted by 2-3 percentage points, top European favourites (FRA, ESP, BRA) commensurately reduced. ARG is the notable exception, upweighted +6.14 pp because its balanced-draw path runs through a softer Conmebol-heavy quarter. At simulator scales (15 qubits) we make no claim of quantum advantage. The contribution is conceptual: the encoding captures bracket structural correlations that classical Monte Carlo only reproduces via explicit path-by-path conditioning, and the construction scales tractably toward the 50-100 logical qubits expected from early fault-tolerant hardware. A French translation is included as an additional file (main_fr.pdf). Source code (Qiskit 2.4 / Python 3.10) and LaTeX sources are bundled in arxiv_upload.tar.gz.