[Paper Review] The quantum low-rank approximation problem

Nic Ezzell, Zoë Holmes+1 more|arXiv (Cornell University)|Mar 2, 2022

Sparse and Compressive Sensing Techniques4 citations

TL;DR

This paper formulates the quantum low-rank approximation problem (QLRAP) for density matrices, solving for the optimal low-rank quantum state $σ$ that minimizes either the Hilbert-Schmidt or trace distance to a given state $\rho$. It shows that the Hilbert-Schmidt distance yields a unique optimal solution $\sigma^* = \tau_R + N_R$, where $\tau_R$ is the projection of $\rho$ onto its $R$ principal components, enabling reliable principal component analysis (PCA) on near-term quantum computers via variational optimization.

ABSTRACT

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 $N_R = \frac{1- ext{Tr}(τ_R)}{R}Π_R$. 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.

Motivation & Objective

• To formalize the quantum low-rank approximation problem (QLRAP) as minimizing distance between a target quantum state $\rho$ and a low-rank state $\sigma$ with $\rank(\sigma) \leq R$.
• To analyze the optimal solution under two distance measures: Hilbert-Schmidt and trace distance.
• To determine which distance metric yields a unique, physically meaningful solution suitable for quantum principal component analysis (PCA).
• To enable variational quantum algorithms for PCA by identifying a cost function that reliably extracts principal components.

Proposed method

• Formulate the QLRAP as minimizing $D_{HS}(\rho, \sigma) = \|\rho - \sigma\|_F^2$ or $D_{\operatorname{tr}}(\rho, \sigma) = \|\rho - \sigma\|_1$ subject to $\rank(\sigma) \leq R$.
• Derive the optimal $\sigma^*$ analytically using spectral decomposition: $\sigma^* = \tau_R + N_R$, where $\tau_R = \Pi_R \rho \Pi_R$ and $N_R = \frac{1 - \Tr(\tau_R)}{R} \Pi_R$.
• Show that the Hilbert-Schmidt solution is unique and preserves eigenvalue ordering, while the trace distance solution is degenerate and may not preserve ordering.
• Propose using the Hilbert-Schmidt distance as a cost function in variational quantum algorithms to extract principal components via optimization.
• Enable efficient estimation of $D_{HS}(\rho, \sigma)$ on quantum hardware using the destructive SWAP test to measure $\Tr(\rho^2)$, $\Tr(\sigma^2)$, and $\Tr(\rho\sigma)$.
• Design variational circuits that either prepare a purification of $\sigma$ or use classical randomness to prepare its eigenvectors, enabling extraction of principal components after optimization.

Experimental results

Research questions

• RQ1 What is the optimal low-rank quantum state $\sigma$ that minimizes the Hilbert-Schmidt distance to a given density matrix $\rho$ under the rank constraint $\rank(\sigma) \leq R$?
• RQ2 How does the solution to the quantum low-rank approximation problem differ when using the trace distance instead of the Hilbert-Schmidt distance?
• RQ3 Can the Hilbert-Schmidt distance be used as a reliable cost function in variational quantum algorithms for quantum principal component analysis (PCA)?
• RQ4 Does the optimal solution preserve the eigenvalue ordering of the original state $\rho$, and why does this matter for PCA?
• RQ5 What are the implications of solution degeneracy in the trace distance case for quantum data compression and quantum machine learning?

Key findings

• The optimal state minimizing the Hilbert-Schmidt distance is uniquely given by $\sigma^* = \tau_R + N_R$, where $\tau_R = \Pi_R \rho \Pi_R$ is the projection of $\rho$ onto its $R$ largest eigenmodes.
• The Hilbert-Schmidt solution preserves the eigenvalue ordering of $\rho$, ensuring correct identification of principal components.
• The trace distance solution is not unique; a degenerate family of states achieves the minimum distance, potentially leading to incorrect eigenvalue ordering.
• The Hilbert-Schmidt distance is more suitable for quantum PCA than the trace distance due to its unique, ordered solution and efficient quantum estimation.
• The optimal state $\sigma^*$ can be variationally prepared on near-term quantum hardware using a cost function based on $D_{HS}(\rho, \sigma)$, enabling resource-efficient PCA.
• The solution enables efficient compression of quantum states by approximating high-rank $\rho$ with low-rank $\sigma^*$, reducing qubit and circuit resource requirements.