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Adaptivity can help exponentially for shadow tomography
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In recent years there has been significant interest in understanding the statistical complexity of learning from quantum data under the constraint that one can only make unentangled measurements. While a key challenge in establishing tight lower bounds in this setting is to deal with the fact that the measurements can be chosen in an adaptive fashion, a recurring theme has been that adaptivity offers little advantage over more straightforward, nonadaptive protocols. In this note, we offer a counterpoint to this. We show that for the basic task of shadow tomography, protocols that use adaptively chosen two-copy measurements can be exponentially more sample-efficient than any protocol that uses nonadaptive two-copy measurements.
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Lower Bounds on Relative Error Quantum Compression and Classical Shadows
The claimed Ω(√(2^n)ε^{-2}) lower bounds for relative-error quantum state compression are not established because the reductions' error propagation is quantitatively invalid.
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