  Quantum speed limit (QSL) is the study of fundamental limits on the evolution
time of quantum systems. For instance, under the action of a time-independent
Hamiltonian, the evolution time between an initial and a final quantum state
obeys various mutually complementary lower bounds. They include the
Mandelstam-Tamm, Margolus-Levitin, Luo-Zhang, dual ML and Lee-Chau bounds. Here
we show that the Mandelstam-Tamm bound can be obtained by optimizing the
Lee-Chau bound over a certain parameter. More importantly, we report a QSL that
includes all the above bounds as special cases before optimizing over the
physically meaningless reference energy level of a quantum system. This
unifying bound depends on a certain parameter $p$. For any fixed $p$, we find
all pairs of time-independent Hamiltonian and initial pure quantum state that
saturate this unifying bound. More importantly, these pairs allow us to compute
this bound accurately and efficiently using an oracle that returns certain
$p$th moments related to the absolute value of energy of the quantum state.
Moreover, this oracle can be simulated by a computationally efficient and
accurate algorithm for finite-dimensional quantum systems as well as for
certain infinite-dimensional quantum states with bounded and continuous energy
spectra. This makes our computational method feasible in a lot of practical
situations. We compare the performance of this bound for the case of a fixed
$p$ as well as the case of optimizing over $p$ with existing QSLs. We find that
if the dimension of the underlying Hilbert space is $\lesssim 2000$, our
unifying bound optimized over $p$ can be computed accurately in a few minutes
using Mathematica code with just-in-time compilation in a typical desktop.
Besides, this optimized unifying QSL is at least as good as all the existing
ones combined and can occasionally be a few percent to a few times better.
