Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

Quantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity, and Efficient Estimators



Yi-Kai Liu, Steven T. Flammia, David Gross, Jens Eisert


Intuitively, if a density operator has small rank, then it should be easier to estimate from experimental data, since in this case only a few eigenvectors need to be learned. We prove two complementary results that confi rm this intuition. First, we show that a low-rank density matrix can be estimated using fewer copies of the state, i.e., the sample complexity of tomography decreases with the rank. Second, we show that unknown low-rank states can be reconstructed from an incomplete set of measurements, using techniques from compressed sensing and matrix completion. These techniques use simple Pauli measurements, and their output can be certifi ed without making any assumptions about the unknown state. We give a new theoretical analysis of compressed tomography, based on the restricted isometry property (RIP) for low-rank matrices. Using these tools, we obtain near-optimal error bounds, for the realistic situation where the data contains noise due to finite statistics, and the density matrix is full-rank with decaying eigenvalues. We also obtain upper-bounds on the sample complexity of compressed tomography, and almost-matching lower bounds on the sample complexity of any procedure using adaptive sequences of Pauli measurements. Using numerical simulations, we compare the performance of two compressed sensing estimators -- the matrix Dantzig selector and the matrix Lasso -- with standard maximum-likelihood estimation (MLE). We nd that, given comparable experimental resources, the compressed sensing estimators consistently produce higher- fidelity state reconstructions than MLE. In addition, the use of an incomplete set of measurements leads to faster classical processing with no loss of accuracy. Finally, we show how to certify the accuracy of a low rank estimate using direct fi delity estimation and we describe a method for compressed quantum process tomography that works for processes with small Kraus rank.
New Journal of Physics


Quantum state tomography, compressed sensing, matrix completion, trace heuristic, matrix Lasso


Liu, Y. , Flammia, S. , Gross, D. and Eisert, J. (2012), Quantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity, and Efficient Estimators, New Journal of Physics (Accessed May 29, 2024)


If you have any questions about this publication or are having problems accessing it, please contact

Created September 27, 2012, Updated February 19, 2017