In this talk, we present a quantum algorithm for numerically solving the Fokker-Planck equation that describes the stochastic processes in cosmology. In particular, we consider the discretization in which time evolution is represented via (sub-)stochastic transition matrices. Based on this discretization, the computation of time evolution of the numerical solution to the Fokker-Planck equation becomes repeated multiplication of the transition matrices, suggesting the possibility of an implementation via quantum algorithms. We present a method to implement this repeated matrix multiplication using a quantum matrix inversion algorithm [1] based on Quantum Singular Value Transformation (QSVT) [2]. As an application, we demonstrate how this method can be used to analyze the phase transition to eternal inflation in cosmic inflation [3].
[1] J. M. Martyn, Z. M. Rossi, A. K. Tan, and I. L. Chuang, “Grand Unification of Quantum Algorithms,” PRX Quantum 2(4), 040203 (2021). https://doi.org/10.1103/PRXQuantum.2.040203
[2] A. Gilyén, Y. Su, G. H. Low, and N. Wiebe, “Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetic,” in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (2019), pp. 193–204. arXiv:1806.01838.
[3] P. Creminelli, S. Dubovsky, A. Nicolis, L. Senatore, and M. Zaldarriaga, “The phase transition to eternal inflation,” J. High Energy Phys. 09 (2008) 036. https://doi.org/10.1088/1126-6708/2008/09/036