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Pauli Center for Theoretical Studies

The Swiss National Science Foundation

ETH Zurich (Computer Science and Physics Department)

Quantum Science and Technology

CQT Singapore

QAP European Project

Sandia National Laboratories

Institute for Quantum Computing

id Quantique

Non-commutative compressed sensing: theory and applications for quantum tomography

David Gross, Hannover

joint work with Yi-Kai Liu, Steven Flammia, Stephen Becker, and Jens Eisert

We establish novel methods for quantum state and process tomography based on compressed sensing. Our protocols require only simple Pauli measurements, and use fast classical post-processing based on convex optimization. Using these techniques, it is possible to reconstruct an unknown density matrix of rank r using O(r d log d) measurement settings, a significant improvement over standard methods that require d² settings. The protocols are stable against noise, and extend to states which are approximately low-rank. The acquired data can be used to certify that the state is indeed close to a low-rank one, so no a priori assumptions are needed.  We present both theoretical bounds and numerical simulations.

At the same time, new mathematical methods for analyzing the problem of low-rank matrix recovery have been obtained.  The methods are both considerably simpler, and more general than previous approaches. It is shown that an unknown d x d matrix of rank r can be efficiently reconstructed given knowledge of only O(d r ν log²d) randomly sampled expansion coefficients with respect to any given matrix basis. The number ν quantifies th e "degree of incoherence'' between the  unknown matrix and the basis. Existing work concentrated mostly on the problem of "matrix completion'', where one aims to recover a low-rank matrix from randomly selected matrix elements.  Our result covers this situation as a special case. The proof consists of a series of relatively elementary steps, which stands in contrast to the highly involved methods previously employed to obtain comparable results.  We discuss operator bases which are incoherent to all low-rank matrices simultaneously. For these bases, we show that O(d r ν log d) randomly sampled expansion coefficients suffice to recover any low-rank matrix with high probability.