  The Time-Fractional Schr\"odinger Equation (TFSE) is well-adjusted to study a
quantum system interacting with its dissipative environment. The Quantum Speed
Limit (QSL) time captures the shortest time required for a quantum system to
evolve between two states, which is significant for evaluating the maximum
speed in quantum processes. In this work, we solve exactly for a generic
time-fractional single qubit open system by applying the TFSE to a basic open
quantum system model, namely the resonant dissipative Jaynes-Cummings (JC)
model, and investigate the QSL time for the system. It is shown that the
non-Markovian memory effects of the environment can accelerate the
time-fractional quantum evolution, thus resulting in a smaller QSL time.
Additionally, the condition for the acceleration evolution of the
time-fractional open quantum system at a given driving time, i.e., a tradeoff
among the fractional order, coupling strength, and photon number, is brought to
light. In particular, a method to manipulate the non-Markovian dissipative
dynamics of a time-fractional open quantum system by adjusting the fractional
order for a long driving time is presented.
