An infinite hierarchy of multi-copy quantum learning tasks
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912638679646208 |
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| author | Nöller, Jan Tran, Viet T. Gachechiladze, Mariami Kueng, Richard |
| author_facet | Nöller, Jan Tran, Viet T. Gachechiladze, Mariami Kueng, Richard |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08070 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An infinite hierarchy of multi-copy quantum learning tasks Nöller, Jan Tran, Viet T. Gachechiladze, Mariami Kueng, Richard Quantum Physics 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. |
| title | An infinite hierarchy of multi-copy quantum learning tasks |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2510.08070 |