Shadow Tomography of Quantum States
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We introduce the problem of *shadow tomography*: given an unknown D-dimensional quantum mixed state ρ, as well as known two-outcome measurements E_1,...,E_M, estimate the probability that E_i 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^-5·^4 M· 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 n^O( 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, Brandão et al. have given a different approach to shadow tomography using semidefinite programming, which achieves a savings in computation time.
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