An Improved Sample Complexity Lower Bound for (Fidelity) Quantum State Tomography

Henry Yuen

doi:10.22331/q-2023-01-03-890

An Improved Sample Complexity Lower Bound for (Fidelity) Quantum State Tomography

Henry Yuen

Columbia University

Published:	2023-01-03, volume 7, page 890
Eprint:	arXiv:2206.11185v2
Doi:	https://doi.org/10.22331/q-2023-01-03-890
Citation:	Quantum 7, 890 (2023).

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Abstract

We show that $\Omega(rd/\epsilon)$ copies of an unknown rank-$r$, dimension-$d$ quantum mixed state are necessary in order to learn a classical description with $1 – \epsilon$ fidelity. This improves upon the tomography lower bounds obtained by Haah, et al. and Wright (when closeness is measured with respect to the fidelity function).

Featured image: Quantum state tomography.

Popular summary

This paper presents a sharper lower bound on the number of copies of a quantum state needed to learn a classical description of it.

► BibTeX data

@article{Yuen2023improvedsample,
  doi = {10.22331/q-2023-01-03-890},
  url = {https://doi.org/10.22331/q-2023-01-03-890},
  title = {An {I}mproved {S}ample {C}omplexity {L}ower {B}ound for ({F}idelity) {Q}uantum {S}tate {T}omography},
  author = {Yuen, Henry},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {890},
  month = jan,
  year = {2023}
}

► References

[1] Dagmar Bruß and Chiara Macchiavello. Optimal state estimation for $d$-dimensional quantum systems. Physics Letters A, 253 (5-6): 249–251, 1999. https://doi.org/10.1016/S0375-9601(99)00099-7.
https://doi.org/10.1016/S0375-9601(99)00099-7

[2] Jeongwan Haah, Aram W Harrow, Zhengfeng Ji, Xiaodi Wu, and Nengkun Yu. Sample-optimal tomography of quantum states. IEEE Transactions on Information Theory, 63 (9): 5628–5641, 2017. https://doi.org/10.1145/2897518.2897585.
https://doi.org/10.1145/2897518.2897585

[3] Michael Keyl and Reinhard F Werner. Optimal cloning of pure states, testing single clones. Journal of Mathematical Physics, 40 (7): 3283–3299, 1999. https://doi.org/10.1063/1.532887.
https://doi.org/10.1063/1.532887

[4] Ryan O'Donnell and John Wright. Efficient quantum tomography. In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing, pages 899–912, 2016. https://doi.org/10.1145/2897518.2897544.
https://doi.org/10.1145/2897518.2897544

[5] Reinhard F Werner. Optimal cloning of pure states. Physical Review A, 58 (3): 1827, 1998. https://doi.org/10.1103/PhysRevA.58.1827.
https://doi.org/10.1103/PhysRevA.58.1827

[6] Andreas Winter. Coding theorem and strong converse for quantum channels. IEEE Transactions on Information Theory, 45 (7): 2481–2485, 1999. https://doi.org/10.1109/18.796385.
https://doi.org/10.1109/18.796385

[7] John Wright. How to learn a quantum state. PhD thesis, Carnegie Mellon University, 2016.

Cited by

[1] Nic Ezzell, Elliott M. Ball, Aliza U. Siddiqui, Mark M. Wilde, Andrew T. Sornborger, Patrick J. Coles, and Zoë Holmes, "Quantum mixed state compiling", Quantum Science and Technology 8 3, 035001 (2023).

[2] Joran van Apeldoorn, Arjan Cornelissen, András Gilyén, and Giacomo Nannicini, "Quantum tomography using state-preparation unitaries", arXiv:2207.08800, (2022).

[3] Srinivasan Arunachalam, Sergey Bravyi, Arkopal Dutt, and Theodore J. Yoder, "Optimal algorithms for learning quantum phase states", arXiv:2208.07851, (2022).

[4] Ming-Chien Hsu, En-Jui Kuo, Wei-Hsuan Yu, Jian-Feng Cai, and Min-Hsiu Hsieh, "Quantum state tomography via non-convex Riemannian gradient descent", arXiv:2210.04717, (2022).

[5] Anurag Anshu and Srinivasan Arunachalam, "A survey on the complexity of learning quantum states", arXiv:2305.20069, (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2023-06-05 08:36:14). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2023-06-05 08:36:12).

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