We investigate the quantum speed limit (QSL) of a single superconducting
qubit subjected to pure dephasing induced by bistable random telegraph noise
(RTN), a common environmental disturbance in solid state quantum systems. Using
an exactly solvable model, we explore how the interplay between RTN parameters
the switching rate, coupling strength, and initial condition governs the
transition between Markovian and non-Markovian dynamics. A coherence based
measure is employed to quantify non-Markovianity, and a unified quantum speed
limit bound is derived based on relative purity. Our results reveal that in
thermodynamic equilibrium, non-Markovian memory effects significantly reduce
the quantum speed limit time, accelerating quantum evolution through
information backflow. In contrast, under non-equilibrium initial conditions,
the system exhibits purely Markovian behavior regardless of coupling strength,
although strong coupling still leads to speedup via enhanced dephasing. These
findings offer valuable insights for designing fast and noise resilient quantum
protocols by controlling both dynamical parameters and noise initialization.

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