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OP-9451Status unassessedQuantum information

Optimal sample complexity of shadow tomography

Optimal sample complexity of shadow tomography — original question and dated source context.

Historical source; current status unassessedAs of 2024-10-27Source & evidence ↓
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Problem statement

Given mm bounded observables O1,…,OmO_1,\ldots,O_m on a dd-dimensional system and copies of an unknown state ρ\rho, estimate each Tr⁡(Oiρ)\operatorname{Tr}(O_i\rho) to additive error ε\varepsilon. Can the quantum sample complexity match the classical rate O(log⁡(m)/ε2)O(\log(m)/\varepsilon^2), or is there a provable separation? The 2024 source cites an upper bound O(log⁡2(m)log⁡(d)/ε4)O(\log^2(m)\log(d)/\varepsilon^4). See the cited papers for normalization and success-probability conventions.

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Progress & source assessment

Historical source dated 2024-10-27. Current resolution status has not been independently reassessed; bounds and claims in the source context refer to that date.

References
  1. [1]Chen, Cotler, Huang, Li and Tang, Some Open Problems in Quantum Learning, FOCS 2024 workshop, slide 7 link

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