The Keyl-Werner algorithm is not optimal for spectrum estimation

2026-07-29

AI Summary

This work breaks through the long-standing barrier of the Keyl–Werner quantum spectrum estimation algorithm, which had remained unimproved since 2001, by achieving the first eigenvalue spectrum estimation with sample complexity below Θ(d²). The proposed method introduces a novel direction-dependent relative-error quantum state tomography technique, wherein the error in any direction |w⟩ scales proportionally to ⟨w|ρ|w⟩. This enables constant-accuracy estimation in total variation distance using only O(d²·(log log d / log d)²) samples. The result refutes Wright’s (2016) conjectured lower bound on the sample complexity of spectrum estimation and further extends to principal component analysis under Bures distance and χ²-divergence tomography, resolving several longstanding open problems in quantum information theory.

Abstract

We give an algorithm which, given $n = O(d^2 \cdot (\log\log(d)/\log(d))^2)$ copies of $ρ$, estimates the eigenvalues of $ρ$ to constant error in total variation distance. Thus, we can learn the eigenvalues of a quantum state with fewer copies than the $Θ(d^2)$ needed to run full state tomography. This is the first improvement to spectrum estimation over the influential Keyl-Werner algorithm, which uses $n = Θ(d^2)$ copies, thereby resolving a question raised by Keyl and Werner in 2001 and refuting a 2016 conjecture of Wright. Our main technical tool is a new tomography guarantee, where the error of tomography in a particular direction $|w\rangle$ scales with $\langle w | ρ|w\rangle$ for all directions simultaneously. From this stronger "relative-error" bound, we recover better algorithms for principal component analysis in Bures distance and tomography in $χ^2$-divergence as corollaries.

Problem

• spectrum estimation
• quantum state
• Keyl-Werner algorithm
• sample complexity
• eigenvalue estimation

Innovation

• spectrum estimation
• quantum state tomography
• relative-error guarantee
• Keyl-Werner algorithm
• sample complexity