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Quantum computers with hundreds of qubits will be available soon. Unfortunately, high device error-rates pose a significant challenge in using these near-term quantum systems to power real-world applications. Executing a program on existing quantum systems generates both correct and incorrect outcomes, but often, the output distribution is too noisy to distinguish between them. In this paper, we show that erroneous outcomes are not arbitrary but exhibit a well-defined structure when represented in the Hamming space. Our experiments on IBM and Google quantum computers show that the most frequent erroneous outcomes are more likely to be close in the Hamming space to the correct outcome. We exploit this behavior to improve the ability to infer the correct outcome. We propose Hamming Reconstruction (HAMMER), a post-processing technique that leverages the observation of Hamming behavior to reconstruct the noisy output distribution, such that the resulting distribution has higher fidelity. We evaluate HAMMER using experimental data from Google and IBM quantum computers with more than 500 unique quantum circuits and obtain an average improvement of 1.37x in the quality of solution. On Google's publicly available QAOA datasets, we show that HAMMER sharpens the gradients on the cost function landscape.