Sorry, ciljettu, not quite. It is actually quite easy as puzzles go; I just wanted to try making it as I hadn't seen the idea before (or is that b4?)
The Queen's Minefield

Just like trying to solve a maze...go backwards!
To my slight shock I just googled "Chess Maze" and discovered that Bruce Albertson has tried this idea before and calls them mazes...

It's not quite the same, but very similar. In Bruce's mazes it's usually also possible to capture certain pieces (especially in queen mazes, though seldom in bishop mazes), which adds to the longevity and complexity of the puzzle. He even has pawn and king mazes which are quite innovative.
I suppose that, if capturing pieces is strictly forbidden, or merely impossible, it could be called a "minefield" (a sub-genre of the maze, perhaps?).
Regarding your composition - very well done. However there is a dual on the first move, both 1. Qa2 and 1. Qa5 are just as good. This is easy to fix though (for instance, moving the pawn from a6 to a4). Also, you probably want to add to the description that only the white queen can move, not the king (here it does not matter, but at least for future puzzles and to avoid confusion). Finally, there is a matter of efficiency - there are quite a bit of black pieces on the board (no problem there) - however, they are not all actually needed for your intended solution to still be the correct, fastest solution. That is where you, as a composer, can do your thing and enhance the problem overall.
Thanks for sharing!
The Queen's Minefield
For this next offering the white queen has unlimited moves but can only move somewhere a black piece cannot take her. How many of her consecutive moves does it take to win the game?