A lot of quantum algorithms start from the uniform superposition state, and then do some finagling to transform that state into the one they cared about. I was wondering if there is any advantage from starting from another "easy to prepare" state (i.e. O(1) circuit depth), that has a closer total variational distance from the state you wanted to prepare?
For example, if I wish to prepare a Gaussian distribution on N states, starting from the uniform superposition, one would would have to "move" amplitudes from the states closer to 0 and N, onto the states near "the middle" states around N/2.
So to me, a natural question to ask is, does this task become easier if you start from a better initial state that's also very efficiently preparable?
Is there a classification of what states can be prepared with only a constant circuit depth without ancilla?