Concentration of Data Encoding in Parameterized Quantum Circuits

Abstract

Variational quantum algorithms have been acknowledged as the leading strategy to realize near-term quantum advantages in meaningful tasks, including machine learning and optimization. When applied to tasks involving classical data, such algorithms generally begin with data encoding circuits and train quantum neural networks (QNNs) to minimize target functions. Although QNNs have been widely studied to improve these algorithms’ performance on practical tasks, there is a gap in systematically understanding the influence of data encoding on the eventual performance. In this paper, we make progress in filling this gap by considering the common data encoding strategies based on parameterized quantum circuits. We prove that, under reasonable assumptions, the distance between the average encoded state and the maximally mixed state could be explicitly upper-bounded with respect to the width and depth of the encoding circuit. This result in particular implies that the average encoded state will concentrate on the maximally mixed state at an exponential speed on depth. Such concentration seriously limits the capabilities of quantum classifiers, and strictly restricts the distinguishability of encoded states from a quantum information perspective. To support our findings, we numerically verify these results on both synthetic and public data sets. Our results highlight the significance of quantum data encoding and may shed light on the future design of quantum encoding strategies.

Publication
36th Conference on Neural Information Processing Systems (NeurIPS 2022)
Guangxi Li
Guangxi Li
Research Associate (incoming March 2024)

I obtained my BS and MS in Computer Science from University of Electronic Science and Technology of China. I obtained my PhD degree in Computer Science from University of Technology Sydney. My research interests include quantum computing and quantum machine learning.

Xin Wang
Xin Wang
Associate Professor

The main focus of my research is to better understand the limits of information processing with quantum systems and the power of quantum artificial intelligence.