Classical shadows for sample-efficient measurements of gauge-invariant observables
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AbstractClassical shadows provide a versatile framework for estimating many properties of quantum states from repeated, randomly chosen measurements without requiring full quantum state tomography. When prior information is available, such as knowledge of symmetries of states and operators, this knowledge can be exploited to significantly improve sample efficiency. In this work, we develop three classical shadow protocols for $\mathbb{Z}_2$ lattice gauge theory, where a dual formulation enables a rigorous analysis of resource requirements, including both circuit depth and sample complexity. Our approaches can offer exponential improvements in sample complexity over symmetry-agnostic methods, albeit at the cost of increased circuit complexity. While our analysis is restricted to $\mathbb{Z}_2$ lattice gauge theory, our approach offers a blueprint for similar protocols for more general lattice gauge theory models which are currently at the forefront of quantum simulation efforts.Featured image: 1) A quantum state satisfying Gauss' law is prepared on a quantum device. 2) Unitary operations randomizing over the symmetric subspace are applied and the subsequent state is measured. 3) Gauge-invariant observables can be estimated using the collected data. The advantages of using this approach are that for many observables of interest, there is a provably exponential reduction in sample complexity required when compared to standard local randomization schemes. However, the circuit complexity of the randomizing operations can grow, leading to a tradeoff between circuit and sample complexity. Popular summaryFully learning an unknown quantum state requires an exponential number of measurements. Nonetheless, using well-chosen random measurements on copies of an unknown quantum state, one can estimate specific properties of interest much more cleverly. In addition, one often knows some properties of the unknown state of interest. For instance, if the state under consideration is the output of a quantum simulation of a physical theory, we know general features, such as symmetries, of the physically-allowed states. In this paper, we develop random measurement protocol for a class of models, gauge theories, which posses a particular intricate structure and symmetries that one can exploit to drastically improve such learning protocols. Such systems, especially lattice gauge theories, have been at the forefront of advancing quantum simulation capabilities. We introduce three new measurement protocols. Each protocol uses prior knowledge of the symmetry to restrict the randomization, thereby exponentially reducing the number of measurement samples needed compared to symmetry- agnostic approaches. However, this reduction in sample complexity comes at a cost: the randomizing operations on the quantum computer become more complex. For near-to-intermediate term quantum simulators, leveraging symmetry via classical shadows offers a promising route to efficient characterization of gauge-invariant observables, provided the hardware can handle the more demanding randomization circuits required.► BibTeX data@article{Bringewatt2026classicalshadows, doi = {10.22331/q-2026-06-08-2127}, url = {https://doi.org/10.22331/q-2026-06-08-2127}, title = {Classical shadows for sample-efficient measurements of gauge-invariant observables}, author = {Bringewatt, Jacob and Froland, Henry and Elben, Andreas and Mueller, Niklas}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2127}, month = jun, year = {2026} }► References [1] J. Eisert, M. Friesdorf, and C. Gogolin, Nat. Phys. 11, 124 (2015). https://doi.org/10.1038/nphys3215 [2] J. Schachenmayer, L. Pollet, M. Troyer, and A. J. 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The above citations are from SAO/NASA ADS (last updated successfully 2026-06-08 11:35:55). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-06-08 11:35:54: Could not fetch cited-by data for 10.22331/q-2026-06-08-2127 from Crossref. This is normal if the DOI was registered recently.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. AbstractClassical shadows provide a versatile framework for estimating many properties of quantum states from repeated, randomly chosen measurements without requiring full quantum state tomography. When prior information is available, such as knowledge of symmetries of states and operators, this knowledge can be exploited to significantly improve sample efficiency. In this work, we develop three classical shadow protocols for $\mathbb{Z}_2$ lattice gauge theory, where a dual formulation enables a rigorous analysis of resource requirements, including both circuit depth and sample complexity. Our approaches can offer exponential improvements in sample complexity over symmetry-agnostic methods, albeit at the cost of increased circuit complexity. While our analysis is restricted to $\mathbb{Z}_2$ lattice gauge theory, our approach offers a blueprint for similar protocols for more general lattice gauge theory models which are currently at the forefront of quantum simulation efforts.Featured image: 1) A quantum state satisfying Gauss' law is prepared on a quantum device. 2) Unitary operations randomizing over the symmetric subspace are applied and the subsequent state is measured. 3) Gauge-invariant observables can be estimated using the collected data. The advantages of using this approach are that for many observables of interest, there is a provably exponential reduction in sample complexity required when compared to standard local randomization schemes. However, the circuit complexity of the randomizing operations can grow, leading to a tradeoff between circuit and sample complexity. Popular summaryFully learning an unknown quantum state requires an exponential number of measurements. Nonetheless, using well-chosen random measurements on copies of an unknown quantum state, one can estimate specific properties of interest much more cleverly. In addition, one often knows some properties of the unknown state of interest. For instance, if the state under consideration is the output of a quantum simulation of a physical theory, we know general features, such as symmetries, of the physically-allowed states. In this paper, we develop random measurement protocol for a class of models, gauge theories, which posses a particular intricate structure and symmetries that one can exploit to drastically improve such learning protocols. Such systems, especially lattice gauge theories, have been at the forefront of advancing quantum simulation capabilities. We introduce three new measurement protocols. Each protocol uses prior knowledge of the symmetry to restrict the randomization, thereby exponentially reducing the number of measurement samples needed compared to symmetry- agnostic approaches. However, this reduction in sample complexity comes at a cost: the randomizing operations on the quantum computer become more complex. For near-to-intermediate term quantum simulators, leveraging symmetry via classical shadows offers a promising route to efficient characterization of gauge-invariant observables, provided the hardware can handle the more demanding randomization circuits required.► BibTeX data@article{Bringewatt2026classicalshadows, doi = {10.22331/q-2026-06-08-2127}, url = {https://doi.org/10.22331/q-2026-06-08-2127}, title = {Classical shadows for sample-efficient measurements of gauge-invariant observables}, author = {Bringewatt, Jacob and Froland, Henry and Elben, Andreas and Mueller, Niklas}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2127}, month = jun, year = {2026} }► References [1] J. Eisert, M. Friesdorf, and C. Gogolin, Nat. Phys. 11, 124 (2015). https://doi.org/10.1038/nphys3215 [2] J. Schachenmayer, L. Pollet, M. Troyer, and A. J. 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