Both 'equate with' and 'equate to' are correct and commonly used in English. They can be used interchangeably in most contexts, but 'equate with' is more commonly used when comparing things that are similar or equivalent, while 'equate to' is often used when assigning a value or making a comparison.

We use 'equate with ' when comparing people/places/etc. with anything else. We use 'equate to ' when we want to say that something or someone is the same as someone or something else. In your example, there is no way that Persia = luxury: one is a place and the other an abstract noun, therefore you cannot equate to.