  We analyze state preparation within a restricted space of local control
parameters between adiabatically connected states of control Hamiltonians. We
formulate a conjecture that the time integral of energy fluctuations over the
protocol duration is bounded from below by the geodesic length set by the
quantum geometric tensor. The conjecture implies a geometric lower bound for
the quantum speed limit (QSL). We prove the conjecture for arbitrary,
sufficiently slow protocols using adiabatic perturbation theory and show that
the bound is saturated by geodesic protocols, which keep the energy variance
constant along the trajectory. Our conjecture implies that any optimal
unit-fidelity protocol, even those that drive the system far from equilibrium,
are fundamentally constrained by the quantum geometry of adiabatic evolution.
When the control space includes all possible couplings, spanning the full
Hilbert space, we recover the well-known Mandelstam-Tamm bound. However, using
only accessible local controls to anneal in complex models such as glasses or
to target individual excited states in quantum chaotic systems, the geometric
bound for the quantum speed limit can be exponentially large in the system size
due to a diverging geodesic length. We validate our conjecture both
analytically by constructing counter-diabatic and fast-forward protocols for a
three-level system, and numerically in nonintegrable spin chains and a nonlocal
SYK model.
