Online learning of quantum states

Scott Aaronson, Elad Hazan, Xinyi Chen, Satyen Kale, Ashwin Nayak

Research output: Contribution to journalConference articlepeer-review

14 Scopus citations


Suppose we have many copies of an unknown n-qubit state . We measure some copies of using a known two-outcome measurement E1, then other copies using a measurement E2, and so on. At each stage t, we generate a current hypothesis !t about the state , using the outcomes of the previous measurements. We show that it is possible to do this in a way that guarantees that |Tr(Ei!t) Tr(Ei)|, the error in our prediction for the next measurement, is at least " at most On/"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(pTn ) 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.

Original languageEnglish (US)
Pages (from-to)8962-8972
Number of pages11
JournalAdvances in Neural Information Processing Systems
StatePublished - 2018
Event32nd Conference on Neural Information Processing Systems, NeurIPS 2018 - Montreal, Canada
Duration: Dec 2 2018Dec 8 2018

All Science Journal Classification (ASJC) codes

  • Computer Networks and Communications
  • Information Systems
  • Signal Processing


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