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Online quantum state learning is a recently proposed problem by Aaronson etal. (2018), where the learner sequentially predicts $n$-qubit quantum statesbased on given measurements on states and noisy outcomes. In the previous work,the algorithms are worst-case optimal in general but fail in achieving tighterbounds in certain simpler or more practical cases. In this paper, we developalgorithms to advance the online learning of quantum states. First, we showthat Regularized Follow-the-Leader (RFTL) method with Tallis-2 entropy canachieve an $O(\sqrt{MT})$ total loss with perfect hindsight on the first $T$measurements with maximum rank $M$. This regret bound depends only on themaximum rank $M$ of measurements rather than the number of qubits, which takesadvantage of low-rank measurements. Second, we propose a parameter-freealgorithm based on a classical adjusting learning rate schedule that canachieve a regret depending on the loss of best states in hindsight, which takesadvantage of low noisy outcomes. Besides these more adaptive bounds, we alsoshow that our RFTL with Tallis-2 entropy algorithm can be implementedefficiently on near-term quantum computing devices, which is not achievable inprevious works.

More Practical and Adaptive Algorithms for Online Quantum State Learning