Definition - There is nothing wrong with starting a sentence with the sentence adverb hopefully.
Note: To read an eloquent defence of this usage, click HERE.

Example -

(1) Hopefully, he'll be back by midnight.
(Translation: I hope he'll be back by midnight.)

Quotation -
"Hopefully at the beginning of a sentence, as a certain cheeky eighth-grader once pointed out to his everlasting social cost, actually functions not as a misplaced modal auxiliary or as a manner adverb like quickly or angrily but as a "sentence adverb" that indicates the speaker's attitude about the state of affairs described by the sentence (examples of perfectly OK sentence adverbs are Clearly, Basically, Luckily), and only SNOOTs educated in the high-pedantic years up to 1960 blindly proscribe it or grade it down."
(Source: From David Foster Wallace's essay "Tense present")

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