An infinite hierarchy of multi-copy quantum learning tasks

Abstract:Learning properties of quantum states from measurement data is a fundamental challenge in quantum information. The sample complexity of such tasks depends crucially on the measurement primitive. While shadow tomography achieves sample-efficient learning by allowing entangling measurements across many copies, it requires prohibitively deep circuits. At the other extreme, two-copy measurements already yield exponential advantages over single-copy strategies in tasks such as Pauli tomography. In this work we show that such sharp separations extend far beyond the two-copy regime: for every prime c we construct explicit learning tasks of degree c, which are exponentially hard with (c - 1)-copy measurements but efficiently solvable with c-copy measurements. Our protocols are not only sample-efficient but also realizable with shallow circuits. Extending further, we show that such finite-degree tasks exist for all square-free integers c, pointing toward a general principle underlying their existence. Together, our results reveal an infinite hierarchy of multi-copy learning problems, uncovering new phase transitions in sample complexity and underscoring the role of reliable quantum memory as a key resource for exponential quantum advantage.

arXiv:2510.08070 [quant-ph]

(or arXiv:2510.08070v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2510.08070 Focus to learn more arXiv-issued DOI via DataCite

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