Quantum Mean-Field Games: Numerical Nash Equilibria via PDE-form HJB Learning

Quantum mean-field games (QMFGs) extend classical mean-field game theory to populations of indirectly observed quantum systems governed by Belavkin stochastic master equations. Prior QMFG literature provides theoretical foundations but, to our knowledge, no computed Hamilton--Jacobi--Bellman (HJB) optimal Nash equilibria. We present two contributions. First, we identify a structural failure mode of trajectory-based discrete Bellman / deep-BSDE training on Belavkin SDEs: when the measurement diffusion direction is aligned with the value-function gradient, the Monte Carlo Bellman residual is dominated by innovation noise, and an asymptotic signal-to-noise analysis shows the Bellman-loss minimizer is limited by an alignment-dependent noise floor that does not vanish as the timestep is refined. We confirm the geometric prediction on two structurally distinct quantum benchmarks: under aligned noise the trained policy matches zero control, while in a near-orthogonal benchmark Bellman training recovers PDE-form (PINN) performance. Second, using PDE-form HJB learning inside a fictitious-play loop, we compute, to our knowledge, the first numerical Nash equilibrium for a multi-type Belavkin-filtered QMFG, exhibiting equilibria for priors $p \in \{0.25, 0.5, 0.75\}$ and proving the symmetric Nash equilibrium at $p=1/2$ by reflection symmetry.