Quantum Analog-Digital Conversion

Many quantum algorithms, such as 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, where as the latter is required to perform arithmetics on a quantum computer. Quantum algorithms like HHL achieve quantum speedups with a sophisticated use of these two encodings. In this work, we present algorithms that converts 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 an 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 amplitude of a quantum state. As an example, we construct a "quantum amplitude perceptron", a quantum version of neural network, and hence has a possible application in the area of quantum machine learning.