Low variance estimations of many observables with tensor networks and informationally-complete measurements

Stefano Mangini; Daniel Cavalcanti

doi:10.22331/q-2025-07-23-1812

Low variance estimations of many observables with tensor networks and informationally-complete measurements

Stefano Mangini^1,2 and Daniel Cavalcanti¹

¹Algorithmiq Ltd, Kanavakatu 3C 00160 Helsinki, Finland.
²QTF Centre of Excellence, Department of Physics, University of Helsinki, P.O. Box 43, FI-00014 Helsinki, Finland.

Published:	2025-07-23, volume 9, page 1812
Editor:	Luca Tagliacozzo
Eprint:	arXiv:2407.02923v3
Doi:	https://doi.org/10.22331/q-2025-07-23-1812
Citation:	Quantum 9, 1812 (2025).

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Abstract

Accurately estimating the properties of quantum systems is a central challenge in quantum computing and quantum information. We propose a method to obtain unbiased estimators of multiple observables with low statistical error by post-processing informationally complete measurements using tensor networks. Compared to other observable estimation protocols based on classical shadows and measurement frames, our approach offers several advantages: (i) it can be optimised to provide lower statistical error, resulting in a reduced measurement budget to achieve a specified estimation precision; (ii) it scales to a large number of qubits due to the tensor network structure; (iii) it can be applied to any measurement protocol with measurement operators that have an efficient tensor-network representation. We benchmark the method through various numerical examples, including spin and chemical systems, and show that our method can provide statistical error that are orders of magnitude lower than the ones given by classical shadows.

Featured image: We propose a method to obtain unbiased estimators of multiple observables with low statistical error by post-processing informationally (over-)complete (IC) measurements using tensor networks. Given a tensor network representation of the observables and the measurement operators (POVM), the procedure works by optimising a tensor network estimator with a DMRG-like procedure to minimise the variance of the prediction.

Numerical experiments on several examples show that our method can achieve statistical errors that are orders of magnitude lower than other observable estimation protocols based on informationally complete measurements, such as classical shadows.

► BibTeX data

@article{Mangini2025lowvariance,
  doi = {10.22331/q-2025-07-23-1812},
  url = {https://doi.org/10.22331/q-2025-07-23-1812},
  title = {Low variance estimations of many observables with tensor networks and informationally-complete measurements},
  author = {Mangini, Stefano and Cavalcanti, Daniel},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1812},
  month = jul,
  year = {2025}
}

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Cited by

[1] Keijo Korhonen, Stefano Mangini, Joonas Malmi, Hetta Vappula, and Daniel Cavalcanti, "Improving shadow estimation with locally optimal dual frames", Physical Review A 114 1, 012428 (2026).

[2] Hyunho Cha, Sangwoo Hong, and Jungwoo Lee, "Operator-aware shadow importance sampling for accurate fidelity estimation", Physical Review A 113 4, 042602 (2026).

[3] Riccardo Cioli, Elisa Ercolessi, Matteo Ippoliti, Xhek Turkeshi, and Lorenzo Piroli, "Approximate inverse measurement channel for shallow shadows", Quantum 9, 1698 (2025).

[4] Keijo Korhonen, Hetta Vappula, Adam Glos, Marco Cattaneo, Zoltán Zimborás, Elsi-Mari Borrelli, Matteo A. C. Rossi, Guillermo García-Pérez, and Daniel Cavalcanti, "Practical techniques for high-precision measurements on near-term quantum hardware and applications in molecular energy estimation", npj Quantum Information 11 1, 110 (2025).

[5] Bujiao Wu, Lingyu Kong, Yuxuan Yan, Fuchuan Wei, and Zhenhuan Liu, "Expectation value estimation with parametrized quantum circuits", arXiv:2407.19499, (2024).

[6] Nathan A. Baker, Brian Bilodeau, Chi Chen, Yingrong Chen, Marco Eckhoff, Alexandra Efimovskaya, Piero Gasparotto, Puck van Gerwen, Rushi Gong, Kevin Hoang, Zahra Hooshmand, Andrew J. Jenkins, Conrad S. N. Johnston, Run R. Li, Jiashu Liang, Hongbin Liu, Alexis Mills, Maximilian Mörchen, George Nishibuchi, Chong Sun, Bill Ticehurst, Matthias Troyer, Jan P. Unsleber, Stefan Wernli, David B. Williams-Young, and Boqin Zhang, "QDK/Chemistry: A Modular Toolkit for Quantum Chemistry Applications", arXiv:2601.15253, (2026).

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