The Keyl-Werner algorithm is not optimal for spectrum estimation

Abstract:We give an algorithm which, given $n = O(d^2 \cdot (\log\log(d)/\log(d))^2)$ copies of $\rho$, estimates the eigenvalues of $\rho$ to constant error in total variation distance. Thus, we can learn the eigenvalues of a quantum state with fewer copies than the $\Theta(d^2)$ needed to run full state tomography. This is the first improvement to spectrum estimation over the influential Keyl-Werner algorithm, which uses $n = \Theta(d^2)$ 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 $\langle w | \rho |w\rangle$ for all directions simultaneously. From this stronger "relative-error" bound, we recover better algorithms for principal component analysis in Bures distance and tomography in $\chi^2$-divergence as corollaries.

Quantum Physics (quant-ph); Computational Complexity (cs.CC); Data Structures and Algorithms (cs.DS)

arXiv:2607.27117 [quant-ph]

(or arXiv:2607.27117v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2607.27117 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.