Investigate options for auto-deleting branches after merging

[mhucka]

Description of the issue

In Cirq Cynq discussions on 2025-01-22, the issue of stale branches in the GitHub Cirq repo was raised. There are now a lot of branches in the repo that are unnecessary to keep around. One of the sources of these branches is pull requests. There are GitHub project settings that affect the behavior around this, and there are also GitHub Actions workflows that can automate things like automatic deletions of branches.

We should investigate the options and implement changes to avoid the proliferation of stale branches.