2022/03/02 by Nic Ezzell, Zoë Holmes, Ezzell, Nic +3 · 1 citation
Computer Science · Engineering · Mathematics · #FOS: Computer and information sciences #FOS: Physical sciences #Machine Learning (cs.LG) #Quantum Computing Algorithms and Architecture #Quantum Physics (quant-ph) #Sparse and Compressive Sensing Techniques #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2203.00811
openalex publication_date 2022/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a quantum version of the famous low-rank approximation problem. Specifically, we consider the distance D(ρ,σ) between two normalized quantum states, ρ and σ, where the rank of σ is constrained to be at most R. For both the trace distance and Hilbert-Schmidt distance, we analytically solve for the optimal state σ that minimizes this distance. For the Hilbert-Schmidt distance, the unique optimal state is σ= τR +NR, where τR = ΠR ρΠR is given by projecting ρ onto its R principal components with projector ΠR, and NR is a normalization factor given by NR = (1- Tr(τR))/(R)ΠR. For the trace distance, this state is also optimal but not uniquely optimal, and we provide the full set of states that are optimal. We briefly discuss how our results have application for performing principal component analysis (PCA) via variational optimization on quantum computers.