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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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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[1] Bujiao Wu and Dax Enshan Koh, "Error-mitigated fermionic classical shadows on noisy quantum devices", npj Quantum Information 10 1, 39 (2024).

[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).

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The above citations are from Crossref's cited-by service (last updated successfully 2024-06-22 14:02:19) and SAO/NASA ADS (last updated successfully 2024-06-22 14:02:19). The list may be incomplete as not all publishers provide suitable and complete citation data.