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Learning quantum states and unitaries of bounded gate complexity
While quantum state tomography is notoriously hard, most states hold little interest to practically-minded tomographers.
A single $T$-gate makes distribution learning hard
We first show that the generative modelling problem associated with depth $d=n^{\Omega(1)}$ local quantum circuits is hard for any learning algorithm, classical or quantum.
Learnability of the output distributions of local quantum circuits
As many practical generative modelling algorithms use statistical queries -- including those for training quantum circuit Born machines -- our result is broadly applicable and strongly limits the possibility of a meaningful quantum advantage for learning the output distributions of local quantum circuits.
Private learning implies quantum stability
We then show information-theoretic implications between DP learning quantum states in the PAC model, learnability of quantum states in the one-way communication model, online learning of quantum states, quantum stability (which is our conceptual contribution), various combinatorial parameters and give further applications to gentle shadow tomography and noisy quantum state learning.
Robust quantum minimum finding with an application to hypothesis selection
We demonstrate a quantum algorithm for noisy quantum minimum-finding that preserves the quadratic speedup of the noiseless case: our algorithm runs in time $\tilde O(\sqrt{N (1+\Delta)})$, where $\Delta$ is an upper-bound on the number of elements within the interval $\alpha$, and outputs a good approximation of the true minimum with high probability.
Adaptive Quantum State Tomography with Neural Networks
We demonstrate that our algorithm learns to work with basis, symmetric informationally complete (SIC), as well as other types of POVMs.
