The quantum complexity of approximating the frequency moments

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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.
Original languageEnglish
Pages (from-to)1169-1190
Number of pages22
JournalQuantum Information and Computation
Issue number13-14
Publication statusPublished - 1 Oct 2016


  • Frequency moments
  • quantum query complexity
  • quantum streaming complexity


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