We introduce the problem of *shadow tomography*: given an unknown
D-dimensional quantum mixed state Ο, as well as known two-outcome
measurements E1β,β¦,EMβ, estimate the probability that Eiβ
accepts Ο, to within additive error Ξ΅, for each of the M
measurements. How many copies of Ο are needed to achieve this, with high
probability? Surprisingly, we give a procedure that solves the problem by
measuring only O(Ξ΅β4β log4Mβ logD) copies. This means, for example, that we can learn the behavior of an
arbitrary n-qubit state, on all accepting/rejecting circuits of some fixed
polynomial size, by measuring only nO(1) copies of the state.
This resolves an open problem of the author, which arose from his work on
private-key quantum money schemes, but which also has applications to quantum
copy-protected software, quantum advice, and quantum one-way communication.
Recently, building on this work, Brand\~ao et al. have given a different
approach to shadow tomography using semidefinite programming, which achieves a
savings in computation time.Comment: 29 pages, extended abstract appeared in Proceedings of STOC'2018,
revised to give slightly better upper bound (1/eps^4 rather than 1/eps^5) and
lower bounds with explicit dependence on the dimension