Quantum Analog-Digital Conversion (1805.11250v2)

Abstract: 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.

Quantum Analog-Digital Conversion: Insights and Developments

The manuscript titled "Quantum Analog-Digital Conversion" authored by Kosuke Mitarai, Masahiro Kitagawa, and Keisuke Fujii presents a significant exploration of the transformation techniques between analog and digital encodings in quantum information. The primary focus is on mechanisms that facilitate interconversion between these forms, which are pivotal in a range of quantum algorithms that promise efficiency advantages over classical computing.

In the field of quantum computation, encoding data into quantum systems can occur via two prominent methods: analog encoding, where data is stored as the amplitudes of quantum states, and digital encoding, where data are encapsulated in a string of qubits. These encoding schemes serve different functions: analog encoding excels in processing data within the vast quantum state space, while digital encoding is necessary for executing arithmetic operations on quantum computers. Notably, algorithms such as the Harrow-Hassidim-Lloyd (HHL) algorithm employ both methods to achieve computational speedups.

This paper offers a detailed formulation of the algorithms for Quantum Digital-to-Analog Conversion (QDAC) and Quantum Analog-to-Digital Conversion (QADC). While digital-to-analog conversion is probabilistic, the authors introduce a generalized QDAC protocol building on methodologies implicit in existing quantum algorithms. Their most notable advancement is a deterministic algorithm for QADC. The prospects of these conversion algorithms are substantial, particularly in enhancing high-level quantum algorithms such as the nonlinear transformation of quantum state amplitudes.

The authors elucidate the theoretical frameworks and operations involved in QDAC and QADC. The digital-encoded data manipulation involves encoded binary strings into quantum amplitude, while the reverse transformation—extracting digital information from quantum amplitudes—is addressed via phase estimation and amplitude amplification techniques. The distinction lies mainly in deterministic versus probabilistic outcomes, timestamping depth analysis, and single- and two-qubit gate requirements.

A significant application discussed is the development of a "quantum amplitude perceptron," a quantum analog of neural network processing. In quantum machine learning, this perceptron can leverage the vast space state operations to enable complex, non-linear computation models. The potential to fuse quantum efficiency with classical machine learning paradigms presents a rich area for subsequent exploration.

While no explicit claims are made regarding the groundbreaking nature of the research, it is prudent to consider the practical and theoretical implications of quantum analog-digital conversion frameworks. Future advancements might define operational boundaries where quantum encoding conversion can overlap with or extend beyond classical paradigms. Another promising direction is refining conversion algorithms to increase the fidelity and success rate, thus paving the way for new, sophisticated quantum applications aimed at data-intensive tasks.

In essence, this research contributes to the foundational understanding of encoding conversions in quantum computing, a crucial aspect for the evolution of quantum data processing frameworks. The theoretical insights provided alongside practical algorithms fortify the quantum computing landscape, opening the door for enhanced algorithmic strategies and computational models in quantum information science.