Performance analysis of multi-shot shadow estimation

You Zhou and Qing Liu

Key Laboratory for Information Science of Electromagnetic Waves (Ministry of Education), Fudan University, Shanghai 200433, China

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Shadow estimation is an efficient method for predicting many observables of a quantum state with a statistical guarantee. In the multi-shot scenario, one performs projective measurement on the sequentially prepared state for $K$ times after the same unitary evolution, and repeats this procedure for $M$ rounds of random sampled unitary. As a result, there are $MK$ times measurements in total. Here we analyze the performance of shadow estimation in this multi-shot scenario, which is characterized by the variance of estimating the expectation value of some observable $O$. We find that in addition to the shadow-norm $\|O \|_{\mathrm{shadow}}$ introduced in [1], the variance is also related to another norm, and we denote it as the cross-shadow-norm $\|O \|_{\mathrm{Xshadow}}$. For both random Pauli and Clifford measurements, we analyze and show the upper bounds of $\|O \|_{\mathrm{Xshadow}}$. In particular, we figure out the exact variance formula for Pauli observable under random Pauli measurements. Our work gives theoretical guidance for the application of multi-shot shadow estimation.

Learning the properties of quantum systems is of fundamental and practical interest.
Shadow estimation is an efficient approach to this task via randomized measurements. To compensate for the cost of the realization of random unitary evolution, one can conduct multi-shot shadow estimation, where projective measurements are performed on the sequentially prepared state for many times under the $same$ unitary evolution, and this procedure is repeated for a few rounds of randomly sampled unitary.

The current work systematically analyzes the statistical performance of shadow estimation in this multi-shot scenario. We introduce a key quantity——the cross-shadow-norm, which supplements the original shadow-norm. For both random Pauli and Clifford measurement settings, we analyze and give the estimation of the cross-shadow-norm. In particular, we figure out the exact variance for Pauli observables under random Pauli measurements and find the advantage of this multi-shot framework. In the meantime, we indicate, with the help of numerical simulation, that the advantage of multi-shot in Clifford measurements is not significant.

Our work gives theoretical guidance for multi-shot shadow estimation, which could be further applied to various quantum learning tasks, from quantum chemistry to quantum many-body physics.

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Cited by

[1] Pei Zeng, "Tailoring randomized-measurement schemes for practical devices", Quantum Views 7, 74 (2023).

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

[3] Jonas Helsen and Michael Walter, "Thrifty Shadow Estimation: Reusing Quantum Circuits and Bounding Tails", Physical Review Letters 131 24, 240602 (2023).

[4] Alessia Suprano, Danilo Zia, Luca Innocenti, Salvatore Lorenzo, Valeria Cimini, Taira Giordani, Ivan Palmisano, Emanuele Polino, Nicolò Spagnolo, Fabio Sciarrino, G. Massimo Palma, Alessandro Ferraro, and Mauro Paternostro, "Experimental Property Reconstruction in a Photonic Quantum Extreme Learning Machine", Physical Review Letters 132 16, 160802 (2024).

[5] Christian Bertoni, Jonas Haferkamp, Marcel Hinsche, Marios Ioannou, Jens Eisert, and Hakop Pashayan, "Shallow shadows: Expectation estimation using low-depth random Clifford circuits", arXiv:2209.12924, (2022).

[6] Benoît Vermersch, Aniket Rath, Bharathan Sundar, Cyril Branciard, John Preskill, and Andreas Elben, "Enhanced Estimation of Quantum Properties with Common Randomized Measurements", PRX Quantum 5 1, 010352 (2024).

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