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Classical shadows with arbitrary group representations
Quantum Physics · Submitted 1 Apr 2026
abstract compiled · 2026-10-15 · no language-model review
The authors' main claim In this work, we present a unified theory that allows us to simultaneously understand CS protocols based on sampling from general group representations, extending previous approaches that worked in simplified (multiplicity-free) settings.
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The abstract, sentence by sentence
Classical shadows (CS) has recently emerged as an important framework to efficiently predict properties of an unknown quantum state. A common strategy in CS protocols is to parametrize the basis in which one measures the state by a random group action; many examples of this have been proposed and studied on a case-by-case basis. In this work, we present a unified theory that allows us to simultaneously understand CS protocols based on sampling from general group representations, extending previous approaches that worked in simplified (multiplicity-free) settings. We identify a class of measurement bases which we call "centralizing bases" that allows us to analytically characterize and invert the measurement channel, minimizing classical post-processing costs. We complement this analysis by deriving general bounds on the sample-complexity necessary to obtain estimates of a given precision. Beyond its unification of previous CS protocols, our method allows us to readily generate new protocols based on other groups, or different representations of previously considered ones. For example, we characterize novel shadow protocols based on sampling from the spin and tensor representations of $\textsf{SU}(2)$, symmetric and orthogonal groups, and the exceptional Lie group $G_2$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Classical shadows (CS) has recently emerged as an important framework to efficiently predict properties of an unknown quantum state. | background 1.00 | 0.04 | 0.03 | asserted 1.00 |
| s1 | A common strategy in CS protocols is to parametrize the basis in which one measures the state by a random group action; many examples of this have been proposed and studied on a case-by-case basis. | background 0.98 | 0.04 | 0.02 | asserted 1.00 |
| s2 | In this work, we present a unified theory that allows us to simultaneously understand CS protocols based on sampling from general group representations, extending previous approaches that worked in simplified (multiplicity-free) settings. | contribution 1.00 | 0.93 | 0.50 | asserted 1.00 |
| s3 | We identify a class of measurement bases which we call "centralizing bases" that allows us to analytically characterize and invert the measurement channel, minimizing classical post-processing costs. | contribution 0.73 | 0.96 | 0.16 | asserted 1.00 |
| s4 | We complement this analysis by deriving general bounds on the sample-complexity necessary to obtain estimates of a given precision. | contribution 0.66 | 0.96 | 0.09 | asserted 1.00 |
| s5 | Beyond its unification of previous CS protocols, our method allows us to readily generate new protocols based on other groups, or different representations of previously considered ones. | contribution 0.73 | 0.89 | 0.23 | asserted 0.92 |
| s6 | For example, we characterize novel shadow protocols based on sampling from the spin and tensor representations of $\textsf{SU}(2)$, symmetric and orthogonal groups, and the exceptional Lie group $G_2$. | contribution 0.70 | 0.94 | 0.11 | asserted 1.00 |
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The paper
Original abstract
Classical shadows (CS) has recently emerged as an important framework to efficiently predict properties of an unknown quantum state. A common strategy in CS protocols is to parametrize the basis in which one measures the state by a random group action; many examples of this have been proposed and studied on a case-by-case basis. In this work, we present a unified theory that allows us to simultaneously understand CS protocols based on sampling from general group representations, extending previous approaches that worked in simplified (multiplicity-free) settings. We identify a class of measurement bases which we call "centralizing bases" that allows us to analytically characterize and invert the measurement channel, minimizing classical post-processing costs. We complement this analysis by deriving general bounds on the sample-complexity necessary to obtain estimates of a given precision. Beyond its unification of previous CS protocols, our method allows us to readily generate new protocols based on other groups, or different representations of previously considered ones. For example, we characterize novel shadow protocols based on sampling from the spin and tensor representations of $\textsf{SU}(2)$, symmetric and orthogonal groups, and the exceptional Lie group $G_2$.
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References
How the literature uses it
5 arXiv papers cite this in 15 sentences
Papers citing it · 8 sentences without a confident label
Cites it as background 3 papers
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We begin by briefly recalling the key points of classical shadows; more details may be found in Appendix [ref] and Refs. [ref] .
Classical shadows over symmetric spaces · Preliminaries · 0.99
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Specifically, second moment computations arise in a wide range of quantum information processing tasks, including quantum advantage experiments [ref] , certification [ref] , randomized benchmarking [ref] , classical shadows [ref] , classical simulation [ref] and concentration phenomena in variational quantum algorithms [ref] .
Boson sampling beyond the dilute regime: second moments and anti-concentration · Introduction · 0.84
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More representation-theoretic formulations for group-based shadows have also been developed, where the measurement channel is analyzed using general group representations [ref] .
Finite-Sample Selected Covariance Spectra in Classical Shadows · Shadow-output vectors and covariance matrices · 0.99
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