  The traditional quantum speed limits are not attainable for many physical
processes, as they tend to be loose and fail to determine the exact time taken
by quantum systems to evolve. To address this, we derive exact quantum speed
limits for the unitary dynamics of pure-state quantum system that outperform
the existing quantum speed limits. Using these exact quantum speed limits, we
can precisely estimate the evolution time for two- and higher-dimensional
quantum systems. Additionally, for both finite- and infinite-dimensional
quantum systems, we derive an improved Mandelstam-Tamm bound for pure states
and show that this bound always saturates for any unitary generated by
self-inverse Hamiltonians. Furthermore, we show that our speed limits establish
an upper bound on the quantum computational circuit complexity. These results
will have a significant impact on our understanding of quantum physics as well
as rapidly developing quantum technologies, such as quantum computing, quantum
control and quantum thermal machines.
