Why does the condition $\langle \varphi _k\left( t \right) |\dot{\varphi}_l\left( t \right) \rangle =0$ called parallel transport condition?

Question

I'm reading this paper (arXiv) about nonadiabatic holonomic quantum computation. Therein, it is stated that $\langle \varphi_k( t ) |\dot{\varphi}_l( t ) \rangle =0$ Eq.(2) is the parallel transport condition, where $|\varphi_k( t ) \rangle $ means quantum state at time $t$ and is the $k$-th eigenstate of the evolution Hamiltonian. I know that parallel transport means moving vectors in manifold parallelly, but I do not quite understand why this formula have the name, and the paper didn't explain it.

Any idea will be helpful, thanks in advance!

Answer 1

Parallel transport is always defined w.r.t. a choice of connection. What I understood from a brief glance is that they don't refer to the metric structure of the quantum state manifold (i.e. Fubini-Study metric and its Levi-Civita connection). That's what you had probably in mind and is displayed in nice pictures like this one. Instead, they are dealing with a more abstract notion of connection, namely a principal connection defined on a frame bundle (similar to gauge theory).

More precisely, they look at the Schrödinger time evolution of a $L$-dimensional subspace under the assumption that it is periodic with period $\tau$. Then, given a basis of this subspace, one can ask how much is this basis rotated after time $\tau$? This is what the concept of holonomy is about and can be quantified by introducing an appropriate connection. Since there's a freedom in choosing the basis, there's a $U(L)$ "gauge freedom". Technically, this means that the connection is a principal connection on a unitary frame bundle of rank $L$. The stated condition is exactly the condition that the time evolution has to fulfill to yield a parallel transport w.r.t. this connection. I suggest to look at previous works, Refs. [13,14] in that paper, to understand that better.