Control cost and quantum speed limit time in controlled almost-exact state transmission in open systems

We investigate the influence of environment noise on control cost and quantum speed limit time (QSLT) in process almost-exact state transmission (AEST) through a spin chain under pulse control. The is immersed its surrounding non-Markovian, finite-temperature heat baths. find that AEST can be realized weak system-bath coupling, low temperature, strong non-Markovian baths effective external Correspondingly, QSLT increase with increasing bath temperature coupling strength. It noticeable non-Markovianity from helpful to reduce shorten QSLT. Furthermore, we there exists trade-off between fidelity higher requires cost. In addition, minimum has been found obtain certain fidelity.