Oh, so by fuzzy search you mean full text search, not just a simple 'LIKE A%' pattern - it's an entirely different kind of flying altogether (https://www.youtube.com/watch?v=3qNtyfZP8bE).

I don't know what kind of internal representation is used to store tsvector column type in PG, but I think GIN index lookups should be very simple to parallelize and combine (I believe a GIN index would basically be a map [word -> set_of_row_ids_that_contain_the_word] so you could perform them on all indexed columns at the same time and then compute intersection of the results?). But maybe two lookups in the same index could be somehow more efficient than two lookups in different indexes, I don't know.

I'm still sceptical about the LIKE/ILIKE scenario though. "WHERE firstname LIKE 'A%' and lastname LIKE 'Smith' " can easily discard all the Joneses and whatnot before even looking at the firstname column, whereas "WHERE combined_firstname_and_lastname LIKE 'A%SMITH' " will only be able to reject "Andrew Jones" after reading the entire string.