Given an unknown D-dimensional quantum state rho, as well as M two-outcome measurements E_1,...,E_M, how many copies of rho do we need, if we want to learn the approximate probability that E_i accepts rho for *every* i?  In this talk, I'll prove the surprising result -- I didn't believe it myself at first -- that one can achieve this using a number of copies that's polylogarithmic in both M and D.  So, e.g., one can learn whether *every* size-n^3 quantum circuit accepts or rejects an n-qubit state, given only poly(n) copies of the state.  To prove this will require first surveying previous results on measuring quantum states and succinctly describing them, including my 2004 postselected learning theorem, and my 2006 "Quantum OR Bound" (with an erroneous proof fixed in 2016 by Harrow, Lin, and Montanaro).

As time permits, I'll also discuss new joint work with Xinyi Chen, Elad Hazan, and Ashwin Nayak, which takes my 2006 result on PAC-learnability of quantum states, and extends to the setting of online learning.  Here we show that, given a sequence of T two-outcome measurements on an n-qubit state, even if the sequence is chosen adversarially, one can still learn to predict the outcomes of thosemmeasurements with total regret O(n) (in the "realizable" case) or O(sqrt(Tn)) (in the "non-realizable" case).

This talk is a part of the University Messenger lecture series, click here for more info. 

Scott Aaronson is the David J. Bruton Centennial Professor of Computer Science at The University of Texas at Austin, and director of its Quantum Information Center. Prior to coming here, he taught for nine years in Electrical Engineering and Computer Science at MIT. His research interests center around the capabilities and limits of quantum computers, and computational complexity theory more generally