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 ($n\geq 2$) 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.

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Cited by

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

[2] 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).

[3] Matteo Ippoliti and Vedika Khemani, "Learnability Transitions in Monitored Quantum Dynamics via Eavesdropper's Classical Shadows", PRX Quantum 5 2, 020304 (2024).

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

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

[6] 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).

[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).

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