Actually, you can persist query plans to disk.
MS SQL allows you to export a query plan to an XML file. You can then alter the query plan as required because contrary to MS documentation, the query optimiser does NOT work out anywhere close to the best execution when multiple view each with many joins are involved.
You can then force the query to execute using your customised plan.
So technically, you can persist a query plan. it's probably not what the question actually meant which was does the query optimiser save plans by default, in which case the answer in no. But you can force a query plan to be used which has been saved to disk.