On the Differential Topology of Expressivity of Parameterized Quantum Circuits

Parameterized quantum circuits play a key role in quantum computing. Measuring the suitability of such a circuit for solving a class of problems is needed. One such promising measure is the expressivity of a circuit, which is defined in two main variants. The variant in focus of this contribution is the so-called dimensional expressivity which measures the dimension of the submanifold of states produced by the circuit. Understanding this measure needs a lot of background from differential topology which makes it hard to comprehend. In this article we provide this background in a vivid as well as pedagogical manner. Especially it strives towards being self-contained for understanding expressivity, e.g. the required mathematical foundations are provided and examples are given. Also, the literature makes several statements about expressivity the proofs of which are omitted or only indicated. In this article we give proofs for key statements from dimensional expressivity, sometimes revealing limits for generalizing them, and also sketching how to proceed in practice to determine this measure.