The preposition across functions to show the direction of a movement from one area (= garden) to another area (=house).
The preposition to is used to show or indicate the destination (of a verb).
Its a short word that points to the destination. (sorry, that is not very technical, but I hope you understand)
The article the is used to refer to a specific place that both the speaker and listener are aware of. (or the house could be the only house in the area).
I hope that helps you.
Some help from The Ins and Outs of Prepositions by Jean Yates Ph.D.