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The infamous numerical sign problem poses a fundamental obstacle toparticle-based stochastic Wigner simulations in high dimensional phase space.Although the existing particle annihilation via uniform mesh significantlyalleviates the sign problem when dimensionality D $\le$ 4, the mesh size growsdramatically when D $\ge$ 6 due to the curse of dimensionality and consequentlymakes the annihilation very inefficient.  In this paper, we propose an adaptive particle annihilation algorithm, termedSequential-clustering Particle Annihilation via Discrepancy Estimation (SPADE),to overcome the sign problem. SPADE follows a divide-and-conquer strategy:Adaptive clustering of particles via controlling their number-theoreticdiscrepancies and independent random matching in each group, and it may learnthe minimal amount of particles that can accurately capture thenon-classicality of the Wigner function. Combining SPADE with the variancereduction technique based on the stationary phase approximation, we attempt tosimulate the proton-electron couplings in 6-D and 12-D phase space. A thoroughperformance benchmark of SPADE is provided with the reference solutions in 6-Dphase space produced by a characteristic-spectral-mixed scheme under a $73^3\times 80^3$ uniform grid, which fully explores the limit of grid-baseddeterministic Wigner solvers.

Overcoming the numerical sign problem in the Wigner dynamics via adaptive particle annihilation