Gallery welcomes head of state's experience (5)
I believe the answer is:
taste
'experience' is the definition.
(taste is a kind of experience)
'gallery welcomes head of state's' is the wordplay.
'gallery' becomes 'tate' (I've seen this before).
'welcomes' is an insertion indicator (in sense of 'invites in').
'head of' indicates taking the first letters.
The initial letter of 'state' is 's'.
'tate' placed around 's' is 'TASTE'.
(Other definitions for taste that I've seen before include "gains experience" , "Sense showing refinement" , "style" , "Liking (for)" , "Perceive the flavour" .)