Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators
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. Firstly, we show that a low-rank density matrix can be estima...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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IOP Publishing
2012-01-01
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| Series: | New Journal of Physics |
| Online Access: | https://doi.org/10.1088/1367-2630/14/9/095022 |
| _version_ | 1827874533419253760 |
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| author | Steven T Flammia David Gross Yi-Kai Liu Jens Eisert |
| author_facet | Steven T Flammia David Gross Yi-Kai Liu Jens Eisert |
| author_sort | Steven T Flammia |
| collection | DOAJ |
| description | 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. Firstly, 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. Secondly, 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. In this paper, we present a new theoretical analysis of compressed tomography, based on the restricted isometry property for low-rank matrices. Using these tools, we obtain near-optimal error bounds for the realistic situation where the data contain 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 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 describe a method for compressed quantum process tomography that works for processes with small Kraus rank and requires only Pauli eigenstate preparations and Pauli measurements. |
| first_indexed | 2024-03-12T16:53:16Z |
| format | Article |
| id | doaj.art-155ae5c0800140a4821ad9cf0ddcb435 |
| institution | Directory Open Access Journal |
| issn | 1367-2630 |
| language | English |
| last_indexed | 2024-03-12T16:53:16Z |
| publishDate | 2012-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | New Journal of Physics |
| spelling | doaj.art-155ae5c0800140a4821ad9cf0ddcb4352023-08-08T11:03:18ZengIOP PublishingNew Journal of Physics1367-26302012-01-0114909502210.1088/1367-2630/14/9/095022Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimatorsSteven T Flammia0David Gross1Yi-Kai Liu2Jens Eisert3Department of Computer Science and Engineering, University of Washington , Seattle, WA, USAInstitute of Physics, University of Freiburg , 79104 Freiburg, GermanyNational Institute of Standards and Technology , Gaithersburg, MD, USADahlem Center for Complex Quantum Systems, Freie Universität Berlin , 14195 Berlin, GermanyIntuitively, 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. Firstly, 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. Secondly, 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. In this paper, we present a new theoretical analysis of compressed tomography, based on the restricted isometry property for low-rank matrices. Using these tools, we obtain near-optimal error bounds for the realistic situation where the data contain 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 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 describe a method for compressed quantum process tomography that works for processes with small Kraus rank and requires only Pauli eigenstate preparations and Pauli measurements.https://doi.org/10.1088/1367-2630/14/9/095022 |
| spellingShingle | Steven T Flammia David Gross Yi-Kai Liu Jens Eisert Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators New Journal of Physics |
| title | Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators |
| title_full | Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators |
| title_fullStr | Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators |
| title_full_unstemmed | Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators |
| title_short | Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators |
| title_sort | quantum tomography via compressed sensing error bounds sample complexity and efficient estimators |
| url | https://doi.org/10.1088/1367-2630/14/9/095022 |
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