For that to make sense, the n-dimensional sphere must not live in an Euclidean n-dimensional space, but a space over the complex numbers (a Hilbert space) -- th

For that to make sense, the n-dimensional sphere must not live in an Euclidean n-dimensional space, but a space over the complex numbers (a Hilbert space) -- that is, coordinates for the space are complex numbers. Also note that (this is a little beside the point) the n-dimensional sphere picture only works for quantum states that can be described with a finite number of dimensions (like in quantum computing): to talk about even simple things like position and momentum of particles, states must live in a infinite-dimensional space. More importantly (as is pointed out in the lecture), every time-evolution of a quantum state from A to B can be represented as a unitary operator, and if you want to take the square root any such operator, you must in general use complex numbers (or something equivalent). Taking the square root like that is a very simple operation that answers the question "what is the operator for the evolution to the point halfway between A and B?"