Universal low-rank matrix recovery from Pauli measurements

TL;DR

This paper demonstrates that low-rank matrices can be efficiently reconstructed from a small number of Pauli measurements using nuclear-norm minimization, with high probability, advancing quantum state tomography techniques.

Contribution

It establishes that almost all sets of O(rd log^6 d) Pauli measurements satisfy the RIP, enabling universal matrix recovery with nearly-optimal error bounds.

Findings

Almost all sets of O(rd log^6 d) Pauli measurements satisfy the RIP.

Low-rank matrices can be recovered from these measurements via nuclear-norm minimization.

The results extend to measurements using orthonormal operator bases with small operator norm.

Abstract

We study the problem of reconstructing an unknown matrix M of rank r and dimension d using O(rd poly log d) Pauli measurements. This has applications in quantum state tomography, and is a non-commutative analogue of a well-known problem in compressed sensing: recovering a sparse vector from a few of its Fourier coefficients. We show that almost all sets of O(rd log^6 d) Pauli measurements satisfy the rank-r restricted isometry property (RIP). This implies that M can be recovered from a fixed ("universal") set of Pauli measurements, using nuclear-norm minimization (e.g., the matrix Lasso), with nearly-optimal bounds on the error. A similar result holds for any class of measurements that use an orthonormal operator basis whose elements have small operator norm. Our proof uses Dudley's inequality for Gaussian processes, together with bounds on covering numbers obtained via entropy duality.