Towards Quantum Tensor Decomposition in Biomedical Applications

TL;DR

The paper surveys tensor decomposition (TD) methods and their biomedical applications, focusing on CP, Tucker, HOSVD, and t-SVD as core techniques across imaging, multi-omics, and spatial transcriptomics, and analyzes fundamental hardness via phase transitions in rank and SNR. It reports a BERTopic-driven literature mapping to identify dominant TD themes and discusses challenges in scalability, rank selection, noise, missing data, and interpretability. The work then introduces a quantum tensor decomposition (QTD) framework, describing how quantum algorithms for spectral tasks (e.g., QPE-based eigenvector recovery) could accelerate TD on multi-modal biomedical data, and provides preliminary resource estimates and a practical pathway for near-term quantum devices. Together, the review and proposed QTD framework map a path toward quantum-accelerated, scalable, and interpretable TD analytics in biomedicine, with clear directions for integrating QC with classical TD approaches.

Abstract

Tensor decomposition has emerged as a powerful framework for feature extraction in multi-modal biomedical data. In this review, we present a comprehensive analysis of tensor decomposition methods such as Tucker, CANDECOMP/PARAFAC, spiked tensor decomposition, etc. and their diverse applications across biomedical domains such as imaging, multi-omics, and spatial transcriptomics. To systematically investigate the literature, we applied a topic modeling-based approach that identifies and groups distinct thematic sub-areas in biomedicine where tensor decomposition has been used, thereby revealing key trends and research directions. We evaluated challenges related to the scalability of latent spaces along with obtaining the optimal rank of the tensor, which often hinder the extraction of meaningful features from increasingly large and complex datasets. Additionally, we discuss recent advances in quantum algorithms for tensor decomposition, exploring how quantum computing can be leveraged to address these challenges. Our study includes a preliminary resource estimation analysis for quantum computing platforms and examines the feasibility of implementing quantum-enhanced tensor decomposition methods on near-term quantum devices. Collectively, this review not only synthesizes current applications and challenges of tensor decomposition in biomedical analyses but also outlines promising quantum computing strategies to enhance its impact on deriving actionable insights from complex biomedical data.

Figures (7)

• Figure 1: CP decomposition of a third-order tensor into $r$ rank-one tensors
• Figure 2: Hierarchy of tensor decompositions. (See Sections \ref{['sec:indscal-to-candelinc']} for mapping of INDSCAL onto CANDELINC, \ref{['sec:dedicom-to-parafac']} for mapping of DEDICOM onto PARAFAC.)
• Figure 3: Tensor decomposition applications in biomedical data in the literature. A. Stacked bar chart of the number of papers from PubMed published with the topic "tensor decomposition" AND categories listed in the legend in the past decade. B. UMAP of the BERTopic embeddings visualizing the different topic clusters. Each data point is a research paper. Applications of "spike tensors" are highlighted with a star symbol.
• Figure 4: Memory usage (in gigabytes) and wall-clock execution time (in hours) for Tucker decomposition of a random dense tensor of size $N^d$, where $N=100$ and the order $d$ is represented in the x-axis.
• Figure 5: Measure of computational hardness in performing Tucker decomposition on biomedical data: A. Memory usage (in Megabytes) in decomposing 3D MRI data of size ($512 \times 784 \times 912$); B. Wall-clock time (in seconds) of decomposing the 3-way MRI data; C. Memory usage (in Megabytes) in decomposing a 3-way multi-omics tensor of size ($281 \times 10000 \times 100)$; B. Wall-clock time (in seconds) of decomposing the 3-way multi-omics tensor.

Theorems & Definitions (6)

• Definition 1
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