Universal Low-Rank Matrix Recovery from Pauli Measurements

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 to compressed sensing methods for quantum state tomography. It is a non-commutative analogue of a well-studied problem: recovering a sparse vector from a few of its Fourier coefficients.

We give a solution to this problem based on the restricted isometry property (RIP), which improves on previous results using dual certificates. In particular, we show that almost all sets of O(rd log^6 d) Pauli measurements satisfy the rank-r 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. Our proof uses Dudley's inequality for Gaussian processes, together with bounds on covering numbers obtained via entropy duality.

Speaker Bio: Yi-Kai Liu is a staff scientist at the National Institute of Standards and Technology (NIST), in Gaithersburg, Maryland. He works on quantum computation and cryptography. Previously he was a postdoc at UC Berkeley and Caltech. He received his PhD in computer science at UC San Diego in 2007.

Presentation Slides: PDF

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