Shadow Tomography of Quantum States

We introduce the problem of shadow tomography: given an unknown D-dimensional quantum mixed state ρ, as well as known two-outcome measurements E₁,…,EM, estimate the probability that Eᵢ accepts ρ, to within additive error ε, for each of the M measurements. How many copies of ρ are needed to achieve this, with high probability? Surprisingly, we give a procedure that solves the problem by measuring only O(ε⁻⁴·log⁴M·log D) 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, Branda͂o et al. have given a different approach to shadow tomography using semidefinite programming, which achieves a savings in computation time.