I have never been to Zimbabwe, but in general (my personal opinion only), they are unlikely to find out about about your conviction unless you inform them yourself.
People with convictions regularly get into developed nations with advanced criminal records/intelligence/computer systems (like the US, Australia, Canada, etc.) by ticking "no" to any questions about convictions. As long as a Zimbabwe visa does not require you to provide a police certificate, you should be OK. I am not a lawyer and this is my personal intuition/experience only, having travelled a fair amount with an unspent conviction and never declaring it.
I would also advise undertaking any embassy enquiries anonymously or using a fake name.