TY - JOUR
T1 - Quantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity, and Efficient Estimators
JF - New Journal of Physics
Y1 - 2012
A1 - Steven T. Flammia
A1 - David Gross
A1 - Yi-Kai Liu
A1 - Jens Eisert
AB - 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 confirm 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 certified 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 with standard maximum-likelihood estimation (MLE). We find 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 fidelity estimation and we describe a method for compressed quantum process tomography that works for processes with small Kraus rank.
VL - 14
U4 - 095022
UR - http://arxiv.org/abs/1205.2300v2
CP - 9
J1 - New J. Phys.
U5 - 10.1088/1367-2630/14/9/095022
ER -
TY - JOUR
T1 - Continuous-variable quantum compressed sensing
Y1 - 2011
A1 - Matthias Ohliger
A1 - Vincent Nesme
A1 - David Gross
A1 - Yi-Kai Liu
A1 - Jens Eisert
AB - We significantly extend recently developed methods to faithfully reconstruct unknown quantum states that are approximately low-rank, using only a few measurement settings. Our new method is general enough to allow for measurements from a continuous family, and is also applicable to continuous-variable states. As a technical result, this work generalizes quantum compressed sensing to the situation where the measured observables are taken from a so-called tight frame (rather than an orthonormal basis) --- hence covering most realistic measurement scenarios. As an application, we discuss the reconstruction of quantum states of light from homodyne detection and other types of measurements, and we present simulations that show the advantage of the proposed compressed sensing technique over present methods. Finally, we introduce a method to construct a certificate which guarantees the success of the reconstruction with no assumption on the state, and we show how slightly more measurements give rise to "universal" state reconstruction that is highly robust to noise.
UR - http://arxiv.org/abs/1111.0853v3
ER -
TY - JOUR
T1 - Quantum state tomography via compressed sensing
JF - Physical Review Letters
Y1 - 2010
A1 - David Gross
A1 - Yi-Kai Liu
A1 - Steven T. Flammia
A1 - Stephen Becker
A1 - Jens Eisert
AB - We establish methods for quantum state tomography based on compressed sensing. These methods are specialized for quantum states that are fairly pure, and they offer a significant performance improvement on large quantum systems. In particular, they are able to reconstruct an unknown density matrix of dimension d and rank r using O(rd log^2 d) measurement settings, compared to standard methods that require d^2 settings. Our methods have several features that make them amenable to experimental implementation: they require only simple Pauli measurements, use fast convex optimization, are stable against noise, and can be applied to states that are only approximately low-rank. The acquired data can be used to certify that the state is indeed close to pure, so no a priori assumptions are needed. We present both theoretical bounds and numerical simulations.
VL - 105
UR - http://arxiv.org/abs/0909.3304v4
CP - 15
J1 - Phys. Rev. Lett.
U5 - 10.1103/PhysRevLett.105.150401
ER -