Diameter of the space of unitary operation manifold for quantum computation?
I am considering the unitary operation manifold for quantum computation. In order to examine the computational complexity of an algorithm using n qubits, we need to define the complexity of a certain unitary operation $U\in SU(2^n)$, where $U$ carries the initial state to the final state of the computation. Usually this is done by define it as the minimal number of simple gates (working on less than 2 qubits) to approximate $U$. <PERSON> shows a Riemannian geometrical description of it where a certain Riemannian metric is defined on $U\in SU(2^n)$ to only count the 'number' of simple gates but put a high penalty on complex gates. Then the complexity problem is understood as finding the shortest geodesic connecting $I$ and $U$ with the given Riemannian metric on $SU(2^n)$. He also indicated that the curvature of this manifold is almost negative everywhere.
My question is: How can I judge the diameter of this manifold? I know usually there are upper bound of the diameter for a manifold with a positive lower curvature bound. 364
But I can not find results on negative curvature manifold. Can I assume that this manifold does not have a finite diameter?
This is very important to understand the complexity of quantum computation and also the complexity of quantum states. We know that every unitary operation $U$ can be APPROXIMATED with an arbitrary accuracy with $e^{n}$ simple gates. For a quantum state, we can also define the complexity of the state as the minimal number of simple operations to achieve it from a simple initial state, for example a product state 000...0000. So this complexity should not be 'approximated' as in quantum computation since the word 'approximation' itself involves the definition of distance. My idea is that if the diameter of the above mentioned manifold is not finite, then we have states with an infinite complexity. If this is the case, and if also this is true for typical states of the state space, then this will change our understanding of quantum system since currently we think the maximal state complexity is $e^n$. 298