Predicting features of complex, large-scale quantum systems is essential to the characterization and engineering of quantum architectures. We present an efficient approach for predicting a large number of linear features using classical shadows obtained from very few quantum measurements. This approach is guaranteed to accurately predict @math linear functions with bounded Hilbert-Schmidt norm from only @math measurement repetitions. This sampling rate is completely independent of the system size and saturates fundamental lower bounds from information theory. We support our theoretical findings with numerical experiments over a wide range of problem sizes (2 to 162 qubits). These highlight advantages compared to existing machine learning approaches.
We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with @math constraint matrices, each of dimension @math , rank @math , and sparsity @math . The first algorithm assumes an input model where one is given access to entries of the matrices at unit cost. We show that it has run time @math , where @math is the error. This gives an optimal dependence in terms of @math and quadratic improvement over previous quantum algorithms when @math . The second algorithm assumes a fully quantum input model in which the matrices are given as quantum states. We show that its run time is @math , with @math an upper bound on the trace-norm of all input matrices. In particular the complexity depends only poly-logarithmically in @math and polynomially in @math . We apply the second SDP solver to the problem of learning a good description of a quantum state with respect to a set of measurements: Given @math measurements and copies of an unknown state @math , we show we can find in time @math a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as @math on the @math measurements, up to error @math . The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics. As in previous work, we obtain our algorithm by "quantizing" classical SDP solvers based on the matrix multiplicative weight method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians with a poly-logarithmic dependence on its dimension, which could be of independent interest. 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 e, 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( e−5·log4 M·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, have given a different approach to shadow tomography using semidefinite programming, which achieves a savings in computation time. 