Quantum analog-digital conversion

Many quantum algorithms, such as the Harrow-Hassidim-Lloyd (HHL) algorithm, depend on oracles that efficiently encode classical data into a quantum state. The encoding of the data can be categorized into two types: analog encoding, where the data are stored as amplitudes of a state, and digital encoding, where they are stored as qubit strings. The former has been utilized to process classical data in an exponentially large space of a quantum system, whereas the latter is required to perform arithmetics on a quantum computer. Quantum algorithms such as HHL achieve quantum speedups with a sophisticated use of these two encodings. In this work, we present algorithms that convert these two encodings to one another. While quantum digital-to-analog conversions have implicitly been used in existing quantum algorithms, we reformulate it and give a generalized protocol that works probabilistically. On the other hand, we propose a deterministic algorithm that performs a quantum analog-to-digital conversion. These algorithms can be utilized to realize high-level quantum algorithms such as a nonlinear transformation of amplitudes of a quantum state. As an example, we construct a ``quantum amplitude perceptron,'' a quantum version of the neural network that hence has a possible application in the area of quantum machine learning.