Generalization Error Bound for Quantum Machine Learning in NISQ Era -- A Survey

TL;DR

This survey systematically maps the state of supervised Quantum Machine Learning generalization bounds in the NISQ era, compiling 37 relevant works from an initial 688 across five databases. It highlights how generalization bounds depend on data size $N$, feature dimension $d$, and circuit/model complexity, and discusses measurement complexity, VC bounds, and kernel-based approaches as central themes. The study reveals pervasive use of classical datasets like MNIST and IRIS, significant noise-induced degradation on real hardware, and a platform- and method-diverse landscape (notably IBM/Qiskit and quantum kernels). It calls for a unified theoretical framework, cross-platform reproducibility, and more quantum-native datasets and techniques to bridge theory and practice in noisy quantum environments.

Abstract

Despite the mounting anticipation for the quantum revolution, the success of Quantum Machine Learning (QML) in the Noisy Intermediate-Scale Quantum (NISQ) era hinges on a largely unexplored factor: the generalization error bound, a cornerstone of robust and reliable machine learning models. Current QML research, while exploring novel algorithms and applications extensively, is predominantly situated in the context of noise-free, ideal quantum computers. However, Quantum Circuit (QC) operations in NISQ-era devices are susceptible to various noise sources and errors. In this article, we conduct a Systematic Mapping Study (SMS) to explore the state-of-the-art generalization bound for supervised QML in NISQ-era and analyze the latest practices in the field. Our study systematically summarizes the existing computational platforms with quantum hardware, datasets, optimization techniques, and the common properties of the bounds found in the literature. We further present the performance accuracy of various approaches in classical benchmark datasets like the MNIST and IRIS datasets. The SMS also highlights the limitations and challenges in QML in the NISQ era and discusses future research directions to advance the field. Using a detailed Boolean operators query in five reliable indexers, we collected 544 papers and filtered them to a small set of 37 relevant articles. This filtration was done following the best practice of SMS with well-defined research questions and inclusion and exclusion criteria.

Figures (4)

• Figure 1: The Prisma diagram provides the literature counts at various stages of the SLR process. From the collected paper count of $688$, the filtration steps excluded $144$ papers, and the eligibility steps excluded $507$ papers, resulting in $37$ papers for the analysis.
• Figure 2: a) Bar graph presenting the number of publications per year from 2010 to 2023 among initially extracted papers. It is evident that the number of publications has increased over the years, mainly in the last 5 years. b) Bar graph for the number of papers extracted per year among the final papers.
• Figure 3: a) Figure showing the number of papers extracted from each database source. Google Scholar was the most significant source for this study. b) Bar graph for the number of papers extracted from each publisher among the final papers. We can see that the majority of the papers were published in ArXiv. This graph represents the final papers after the filtration process. Naturally, it is expected that the platform with free access to full-text articles would have the most papers.
• Figure 4: a) Bar graph for a count of authors per paper. For most papers, we can see a collective research effort trend of $2-5$ authors per paper. b) Bar graph for the number of papers per publication type. The majority of the papers were journal articles. This suggests a preference for formal, peer-reviewed research channels over conference papers.