Compression with random dictionary
This is provably impossible. The proof of Shannon's theorem goes through as usual.

As a simple example, suppose that are looking for an algorithm that compresses all $n$-bit strings to smaller strings. This is impossible since there are $2^n$ possible inputs, but only $2^n-1$ possible outputs. The proof of Shannon's theorem is very similar. Note how $Q$ makes absolutely no difference.

What happens if you choose a random 1 MiB window of $Q$ as your file? This can easily be compressed to 30 bits, by sending the location of the window. However, in this case the entropy of $Q$ is indeed at most 30 bits, since there are at most $2^{30}$ different windows.