In this work we provide new techniques for a fundamental and ubiquitous task: simulating measurement of a quantum state in the standard basis. Our algorithms reduce the sampling task to computing poly(n) amplitudes of n-qubit states; unlike previously known techniques they do not require computation of marginal probabilities. First we consider the case where the state of interest is the output state of an m-gate quantum circuit U. We propose an exact sampling algorithm which involves computing O(m) amplitudes of n-qubit states generated by subcircuits of U spanned by the first t=1,2,…,m gates. We show that our algorithm can significantly accelerate quantum circuit simulations based on tensor network contraction methods or low-rank stabilizer decompositions. Second, we consider the case in which ψ is the unique ground state of a local Hamiltonian with a spectral gap that is lower bounded by an inverse polynomial function of n. We prove that a simple Metropolis-Hastings Markov Chain mixes rapidly to the desired probability distribution provided that ψ obeys a certain technical condition, which we show is satisfied for all sign-problem free Hamiltonians. This talk will be mostly based on arXiv:2112.08499 which is joint work with Sergey Bravyi and Yinchen Liu.