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Publication - Dr Ashley Montanaro

    The quantum complexity of approximating the frequency moments


    Montanaro, A, 2016, ‘The quantum complexity of approximating the frequency moments’. Quantum Information and Computation, vol 16., pp. 1169-1190


    The k’th frequency moment of a sequence of integers is defined as Fk = ∑jnkj, where nis the number of times that j occurs in the sequence. Here we study the quantum complexity of approximately computing the frequency moments in two settings. In the query complexity setting, we wish to minimise the number of queries to the input used to approximate Fk up to relative error ε. We give quantum algorithms which outperform the best possible classical algorithms up to quadratically. In the multiple-pass streaming setting, we see the elements
    of the input one at a time, and seek to minimise the amount of storage space, or passes over the data, used to approximate Fk. We describe quantum algorithms for F0, F2 and F in this model which substantially outperform the best possible classical algorithms in certain parameter regimes.

    Full details in the University publications repository