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Contemporary quantum computers have relatively high levels of noise, making it difficult to use them to perform useful calculations, even with a large number of qubits. Quantum error correction is expected to eventually enable fault-tolerant quantum computation at large scales, but until then it will be necessary to use alternative strategies to mitigate the impact of errors. We propose a near-term friendly strategy to mitigate errors by entangling and measuring $M$ copies of a noisy state $ρ$. This enables us to estimate expectation values with respect to a state with dramatically reduced error, $ρ^M/ \mathrm{Tr}(ρ^M)$, without explicitly preparing it, hence the name "virtual distillation". As $M$ increases, this state approaches the closest pure state to $ρ$, exponentially quickly. We analyze the effectiveness of virtual distillation and find that it is governed in many regimes by the behavior of this pure state (corresponding to the dominant eigenvector of $ρ$). We numerically demonstrate that virtual distillation is capable of suppressing errors by multiple orders of magnitude and explain how this effect is enhanced as the system size grows. Finally, we show that this technique can improve the convergence of randomized quantum algorithms, even in the absence of device noise.