An Efficient Algorithm for Extending Uniform Superposition States for Time-Series Data

Leveraging fundamental quantum features such as quantum superposition is one of the key elements for the theoretical speed-up obtained in quantum algorithms, e.g., with Grover's search algorithm. In this work, we explore the novel concept of dynamic state preparation, which allows for extending an initially prepared superposition state already containing a set of data elements. This state is modified upon receiving further time-series measurement data from an experiment so that this data can be processed with a quantum algorithm to extract the global properties of the system efficiently. Specifically, we distinguish the algorithmic case in which the state-preparation unitary of the initially prepared state to be extended is known and the one where it is not. The former can be described as an application of amplitude amplification, whereas the latter includes an amplitude reservoir assigned to a reference state that allows for extension without knowing the full state preparation unitary. We discuss both cases algorithmically and present the respective quantum circuit, including a specific initial state preparation method. Our analysis reveals a polynomial scaling and respective tradeoffs to be considered.