arXiv Machine Learning October 16, 2026 By Yen-Huan Li, Wei-Fu Tseng

Regret Lower Bound for Online Learning of Quantum States with Logarithmic Loss, with Applications to Quantum State Tomography and Confidence Sequences for Matrix Means

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The paper investigates online learning of quantum states using logarithmic loss, a quantum analogue of online portfolio selection. It establishes a regret lower bound, proving that existing algorithms like Exponential Weights and Tseng et al. (2025) achieve optimal polynomial dependence on dimension. Leveraging this lower bound, the authors present a sample‑optimal quantum state tomography algorithm (up to polylogarithmic factors) and introduce a method for constructing confidence sequences for matrix‑valued random variables, with numerical results indicating competitive performance.

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