Yuhan Liu

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In the quest to identify further quantum algorithms exhibiting superpolynomial speed-ups, a recurring theme is that the complexity of a problem is largely shaped by the input access model. Here, we study this phenomenon for the state hidden subgroup problem (StateHSP), a quantum generalization of the hidden subgroup problem in which the goal is to identify the symmetries of an unknown quantum state. We settle the complexity of the abelian StateHSP in both the previously studied sample model and a new query model, which is a stronger and operationally natural generalization that provides access to the state-preparation unitary and its inverse.  In the query model, we give a time-efficient quantum algorithm using $O(\log(|G/H|)/\sqrt{\epsilon})$ forward and inverse queries, and prove a matching $\Omega(\log(|G/H|)/\sqrt{\epsilon})$ lower bound which holds even in the stronger conjugate-query and controlled-query settings. By contrast, we show that in the sample model, $\Theta(\log(|G/H|)/\epsilon)$ copies are both sufficient and information-theoretically necessary. As applications, we obtain faster algorithms for learning stabilizer groups, locating unentanglement, and identifying hidden translation symmetries.

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Max Planck Institute of Quantum Optics
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Contact: Mingru Yang