Krylov and Nielsen complexity
Oleg Evnin (Chulalongkorn University, Bangkok)
SciPost Phys. 13 (2022) 090
SciPost Phys. 16 (2024) 041
Phys. Rev. Lett. 132 (2024) 160402
Phys. Rev. Lett. 134 (2025) 050402
Recent years have seen a surge of interest in developing measures of complexity for the unitary evolution in quantum mechanics. One approach, known as Nielsen complexity, visualizes the evolution as a continuous « program » executed by the system and applies measures developed in computational complexity theory to judge whether the unitary evolution operators are simple or complex. Another approach, known as Krylov complexity, tracks how rapidly quantum states and operators spread over different independent directions as they evolve. One key question is whether these quantities are capable of distinguishing the evolution operators of solvable (simple) systems from generic (chaotic, complicated) ones. I will describe the challenges and achievements within this line of pursuit, as well as interrelations between the two apparently different approaches.