Online learning of quantum states

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

Research output: Contribution to journalArticlepeer-review

18 Scopus citations


Suppose we have many copies of an unknown n-qubit state ρ. We measure some copies of ρ 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 ωt about the state ρ, using the outcomes of the previous measurements. We show that it is possible to do this in a way that guarantees that |Tr (Eiωt) - Tr (Eiρ)|, the error in our prediction for the next measurement, is at least ϵ at most O(n/ϵ2) 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(√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.

Original languageEnglish (US)
Article number124019
JournalJournal of Statistical Mechanics: Theory and Experiment
Issue number12
StatePublished - Dec 20 2019

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • machine learning


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