General rights This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: In this work we use the concept of quantum fingerprinting to develop a quantum communication protocol in the simultaneous message passing model that calculates the Hamming distance between two n-bit strings up to relative error . The number of qubits communicated by the protocol is polynomial in log n and 1/ ,

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... le any classical protocol must communicate ( √ n) bits. Motivated by the relationship between Hamming distance and vertex distance in hypercubes, we apply the protocol to approximately calculate distances between vertices in graphs that can be embedded into a hypercube such that all distances are preserved up to a constant factor. Such graphs are known as 1 -graphs. This class includes all trees, median graphs, Johnson graphs, and Hamming graphs. Our protocol is efficient for 1 -graphs with low diameter, and we show that its dependence on the diameter is essentially optimal. Finally, we show that our protocol can be used to approximately compute 1 distances between vectors efficiently.