IQI Weekly Seminar
Abstract: In the problem of quantum state learning, one is given a small number of "samples" of a quantum state, and the goal is to output a good estimate of the quantum state. This is a problem which is not only of theoretical interest, but is also commonly used in current-day implementation and verification of quantum technologies. In this talk, I will describe the first optimal quantum algorithm for this problem. In addition, I will describe optimal algorithms for the related problems of testing and learning specific properties of quantum states. These results make use of a new connection between quantum state learning and longest increasing subsequences of random words, a famous topic in combinatorics dating back to a 1935 paper of Erdős and Szekeres. Motivated by this connection, I will show new and optimal bounds on the length of the longest increasing subsequence of a random word.
This is based on joint works with Costin Badescu and Ryan O'Donnell.