Toward Mixed Analog-Digital Quantum Signal Processing: Quantum AD/DA Conversion and the Fourier Transform

TL;DR

This paper introduces a new hybrid quantum signal processing paradigm that combines analog and digital quantum systems, enabling quantum analog-digital conversion and novel implementations of quantum algorithms like the Fourier transform.

Contribution

It develops the first framework for mixed analog-digital quantum signal processing, bridging the gap between CV and DV quantum systems for scalable quantum computation.

Findings

Demonstrates quantum analog-digital conversion between CV and DV systems.

Realizes quantum Fourier transform via quantum oscillator evolution.

Highlights exponential runtime in qubits due to information-theoretic limits.

Abstract

Signal processing stands as a pillar of classical computation and modern information technology, applicable to both analog and digital signals. Recently, advancements in quantum information science have suggested that quantum signal processing (QSP) can enable more powerful signal processing capabilities. However, the developments in QSP have primarily leveraged \emph{digital} quantum resources, such as discrete-variable (DV) systems like qubits, rather than \emph{analog} quantum resources, such as continuous-variable (CV) systems like quantum oscillators. Consequently, there remains a gap in understanding how signal processing can be performed on hybrid CV-DV quantum computers. Here we address this gap by developing a new paradigm of mixed analog-digital QSP. We demonstrate the utility of this paradigm by showcasing how it naturally enables analog-digital conversion of quantum signals -- specifically, the transfer of states between DV and CV quantum systems. We then show that such quantum analog-digital conversion enables new implementations of quantum algorithms on CV-DV hardware. This is exemplified by realizing the quantum Fourier transform of a state encoded on qubits via the free-evolution of a quantum oscillator, albeit with a runtime exponential in the number of qubits due to information theoretic arguments. Collectively, this work marks a significant step forward in hybrid CV-DV quantum computation, providing a foundation for scalable analog-digital signal processing on quantum processors.