Performance Analysis Of Multi-shot Shadow Estimation

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

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Abstract

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.

Featured image: Multi-shot shadow estimation: To compensate for the cost of the realization of random unitary evolution, one can conduct projective measurements on the sequentially prepared state for K times under the same unitary evolution, and this procedure is repeated for M rounds of randomly sampled unitary. The current work systematically analyzes the statistical performance of shadow estimation in this multi-shot scenario, as a supplement to the original shadow estimation [Huang et.al. Nat. Phys. 2020] (K=1).

Popular summary

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.

► BibTeX data

@article{Zhou2023performanceanalysis, doi = {10.22331/q-2023-06-29-1044}, url = {https://doi.org/10.22331/q-2023-06-29-1044}, title = {Performance analysis of multi-shot shadow estimation}, author = {Zhou, You and Liu, Qing}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {7}, pages = {1044}, month = jun, year = {2023}<br /> }

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Cited by

[1] 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).

[2] Benoît Vermersch, Aniket Rath, Bharathan Sundar, Cyril Branciard, John Preskill, and Andreas Elben, “Enhanced estimation of quantum properties with common randomized measurements”, arXiv:2304.12292, (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2023-06-29 13:54:32). The list may be incomplete as not all publishers provide suitable and complete citation data.

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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-2023-06-29-1044/

Time Stamp: June 29, 2023

Time Stamp: Oct 10, 2023

Republished By Plato

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