I have been going through Eddie Farhi's 6-pages long pre-Adiabatic paper, An Analog Analogue of a Digital Quantum Computation.
I guess I understand most of the math and physics but I am struggling with the intuitions used throughout the paper. Here are my questions.
Introduction of $|r\rangle$ on page 3: I understand that $|r\rangle$ and $|w\rangle$ are orthonormal. But I don't see why the authors introduced this state.
Use of Norm in equation 17 on page 4: The equation is as follows. $$\sum_w |||\Psi_w, t\rangle - |\Psi, t\rangle||^2 \ge N \varepsilon$$ There are more than one kind of norms. What kind of norm are we talking about here? Why? What is the physical significance of this quantity in this context?