The quantum low-rank approximation problem
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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 = τ_R +N_R, where τ_R = Π_R ρΠ_R is given by projecting ρ onto its R principal components with projector Π_R, and N_R is a normalization factor given by N_R = 1- Tr(τ_R)/RΠ_R. 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.
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