The compile tests (and other jobs) with Python 3.9 are likely going to fail until all of Aer's upstream python deps support python 3.9 and publish wheels with precompiled binaries. I've been tracking this for terra here: Qiskit/qiskit#5189 and was planning to push an Aer PR after that terra PR merges. I expect it'll be a bit longer before the entire ecosystem is supporting 3.9 though (since numpy only released last week with 3.9 support).

Are we getting closer to the release of a wheel for Python 3.9? The build process is a bit slow:
https://github.com/TheAlgorithms/Python/pull/3926/checks?check_run_id=1447346191#step:5:148

Thanks.

#1071 just merged which added python 3.9 and full CI for aer which superseded this PR. So I'm going to close this one now. Please feel free to reopen it if I'm missing something. The next release will include 3.9 support and wheels for all supported platforms.