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Poster
Online Learning of Quantum States
Scott Aaronson · Xinyi Chen · Elad Hazan · Satyen Kale · Ashwin Nayak

Thu Dec 06 07:45 AM -- 09:45 AM (PST) @ Room 517 AB #166
in Thu Poster Session A »
Suppose we have many copies of an unknown n-qubit state $\rho$. We measure some copies of $\rho$ using a known two-outcome measurement E_1, then other copies using a measurement E_2, and so on. At each stage t, we generate a current hypothesis $\omega_t$ about the state $\rho$, using the outcomes of the previous measurements. We show that it is possible to do this in a way that guarantees that $|\trace(E_i \omega_t) - \trace(E_i\rho)|$, the error in our prediction for the next measurement, is at least $eps$ at most $O(n / eps^2) $\ times. Even in the non-realizable setting---where there could be arbitrary noise in the measurement outcomes---we show how to output hypothesis states that incur at most $O(\sqrt {Tn}) $ excess loss over the best possible state on the first $T$ measurements. These results generalize a 2007 theorem by Aaronson on the PAC-learnability of quantum states, to the online and regret-minimization settings. We give three different ways to prove our results---using convex optimization, quantum postselection, and sequential fat-shattering dimension---which have different advantages in terms of parameters and portability.
Keywords: Information Theory · Online Learning
[ Paper] 

Author Information

Scott Aaronson (UT Austin)
Xinyi Chen (Google Brain)
Elad Hazan (Princeton University)
Satyen Kale (Google)
Ashwin Nayak (University of Waterloo)

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