Classical Shadows Based On Locally-entangled Measurements

Matteo Ippoliti

Department of Physics, The University of Texas at Austin, Austin, TX 78712, USA
Department of Physics, Stanford University, Stanford, CA 94305, USA

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Abstract

We study classical shadows protocols based on randomized measurements in $n$-qubit entangled bases, generalizing the random Pauli measurement protocol ($n = 1$). We show that entangled measurements ($ngeq 2$) enable nontrivial and potentially advantageous trade-offs in the sample complexity of learning Pauli expectation values. This is sharply illustrated by shadows based on two-qubit Bell measurements: the scaling of sample complexity with Pauli weight $k$ improves quadratically (from $sim 3^k$ down to $sim 3^{k/2}$) for many operators, while others become impossible to learn. Tuning the amount of entanglement in the measurement bases defines a family of protocols that interpolate between Pauli and Bell shadows, retaining some of the benefits of both. For large $n$, we show that randomized measurements in $n$-qubit GHZ bases further improve the best scaling to $sim (3/2)^k$, albeit on an increasingly restricted set of operators. Despite their simplicity and lower hardware requirements, these protocols can match or outperform recently-introduced “shallow shadows” in some practically-relevant Pauli estimation tasks.

Featured image: Left: a circuit that implements locally-entangled measurements. Right: partitioning of a qubit array into entangled pairs, which makes it easier to learn the expectation value of some operators (green) but not others (red).

► BibTeX data

@article{Ippoliti2024classicalshadows, doi = {10.22331/q-2024-03-21-1293}, url = {https://doi.org/10.22331/q-2024-03-21-1293}, title = {Classical shadows based on locally-entangled measurements}, author = {Ippoliti, Matteo}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {8}, pages = {1293}, month = mar, year = {2024} }

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Cited by

[1] Benoît Vermersch, Marko Ljubotina, J. Ignacio Cirac, Peter Zoller, Maksym Serbyn, and Lorenzo Piroli, “Many-body entropies and entanglement from polynomially-many local measurements”, arXiv:2311.08108, (2023).

[2] Matteo Ippoliti and Vedika Khemani, “Learnability transitions in monitored quantum dynamics via eavesdropper’s classical shadows”, arXiv:2307.15011, (2023).

[3] Bujiao Wu and Dax Enshan Koh, “Error-mitigated fermionic classical shadows on noisy quantum devices”, arXiv:2310.12726, (2023).

[4] Dominik Šafránek and Dario Rosa, “Measuring energy by measuring any other observable”, Physical Review A 108 2, 022208 (2023).

[5] Arkopal Dutt, William Kirby, Rudy Raymond, Charles Hadfield, Sarah Sheldon, Isaac L. Chuang, and Antonio Mezzacapo, “Practical Benchmarking of Randomized Measurement Methods for Quantum Chemistry Hamiltonians”, arXiv:2312.07497, (2023).

[6] Tianren Gu, Xiao Yuan, and Bujiao Wu, “Efficient measurement schemes for bosonic systems”, Quantum Science and Technology 8 4, 045008 (2023).

[7] Yuxuan Du, Yibo Yang, Tongliang Liu, Zhouchen Lin, Bernard Ghanem, and Dacheng Tao, “ShadowNet for Data-Centric Quantum System Learning”, arXiv:2308.11290, (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2024-03-22 10:19:38). 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 2024-03-22 10:19:37).

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.

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Source: https://quantum-journal.org/papers/q-2024-03-21-1293/

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Classical shadows based on locally-entangled measurements

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