I am reading about ligamentum arteriosum which connects the left pulmonary artery and descending aorta. But I am seeing that if any figure shows left brachiochephalic vein it does not show left pulmonary artery (and vice versa). Are they both the same?
No, they are not the same. The left brachiocephalic vein receives de-oxygenated blood from the left subclavian vein (draining the left arm) and the left internal jugular vein (draining the brain and parts of the face). The brachiocephalic vein then joins with the superior vena cava which flows into the right atrium.
The left pulmonary artery carries the deoxygenated blood that has just passed through the right atrium and ventricle, through the pulmonary trunk to the left lung for oxygenation.
In the foetal circulation, the ductus arteroisus (before it closes into the ligamentum arteriosum) allows blood to bypass the lungs, which are not oxygenated in the uterus!
Please see this diagram (which shows the foetal circulation including PDA for you). The left brachiocephalic vein is circled in yellow, whilst the left pulmonary artery is circled in green.
Image from public domain edition of Gray's.