I'm reading this paper about nonadiabatic holonomic quantum computation and met the condition eq.(2) states that $\langle \varphi _k\left( t \right) |\dot{\varphi}_l\left( t \right) \rangle =0$ is the parallel transport condition, where $|\varphi _k\left( t \right) \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!


1 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.