To Örebro University

The unmanageable amount of encoded information in a many-body particles system makes calculations of its dynamic a challenge. Because of the many degrees of correlation between the particle states, the complexity of the many-body state increases exponentially with the number of particles and their available states. A method that can manage this challenge of complexity is the Gaussian Phase-Space Representation (GPSR) [1]. In GPSR, the wave-function is mapped to a density probability of one-particle density matrices, and the time-dependent Schrödingerequation to a Fokker-Planck equation (FPE). The FPE has to be solved with stochastic differential equations but unfortunately, using stochastic processes induces a maximum simulation time, because some trajectories will end up diverging, which nullifies the validity of averages and prevents us from recovering quantum observables. However, there is a freedom in the decomposition of the diffusion matrix D into the noise matrix B, with D = BBT, and we can extend the maximum simulation time by choosing a ‘better’ noise matrix [2, 3]. We present here the results of GPSR with a noise-matrix computed with a random-SVD [4], which doubles the maximum simulation time while keeping the computational cost reasonable. Understanding why this decomposition works so well might help us finding even better decompositions. 

References

[1] Corney, J. F. and Drummond, P. D. Gaussian operator bases for correlated fermions. Journal of Physics A: Mathematical and General, Volume 39, 269 (2005).

[2] Ögren, M., Kheruntsyan, K. V. and Corney, Joel F. Stochastic simulations of fermionic dynamics with phase-space representations. Computer Physics Communications, Volume 182, 1999 (2011).

[3] Rousse, F., Fasi, M., Dmytryshyn, A., Gulliksson, M. and Ögren, M. Simulations of quantum dynamics with fermionic phase-space representations using numerical matrix factorizations as stochastic gauges. Journal of Physics A: Mathematical and Theoretical, Volume 57, 015303, (2024).

[4] Halko, N., Martinsson, P. G. and Tropp, J. A. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM review,Volume 53, 217, (2011).