The hard problem solved by satellites is getting the satellite in orbit and getting it to stay there.
Humanity already could send radio waves across the planet. Emitter and receiver is not the hard problem.
A 1000-qubit QC can't break a RSA-2048 key, let alone a lot of other interesting tasks. Quantum computers aren't magical things that provide exponential speedups on absolutely everything; they can only provide exponential speedups on some algorithms, and those algorithms generally require linear numbers of qubits to the problem size, so 1000 qubits is greatly limiting to problem size.
Neither any classical computer can. We don't even have enough harddrive to store all quantum information in a 100 qubit QC, let alone 1000qubit. QC is limited to solve a subset of problems do not automatically equals to QC is useless comparing to classical ones. Also not able to invalidate the powerfulness of QC beyond 100 qubits.