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Improved reversible and quantum circuits for Karatsuba-based integer multiplication

Authors Alex Parent, Martin Roetteler, Michele Mosca

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Alex Parent
Martin Roetteler
Michele Mosca

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Alex Parent, Martin Roetteler, and Michele Mosca. Improved reversible and quantum circuits for Karatsuba-based integer multiplication. In 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 73, pp. 7:1-7:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


Integer arithmetic is the underpinning of many quantum algorithms, with applications ranging from Shor's algorithm over HHL for matrix inversion to Hamiltonian simulation algorithms. A basic objective is to keep the required resources to implement arithmetic as low as possible. This applies in particular to the number of qubits required in the implementation as for the foreseeable future this number is expected to be small. We present a reversible circuit for integer multiplication that is inspired by Karatsuba's recursive method. The main improvement over circuits that have been previously reported in the literature is an asymptotic reduction of the amount of space required from O(n^1.585) to O(n^1.427). This improvement is obtained in exchange for a small constant increase in the number of operations by a factor less than 2 and a small asymptotic increase in depth for the parallel version. The asymptotic improvement are obtained from analyzing pebble games on complete ternary trees.
  • Quantum algorithms
  • reversible circuits
  • quantum circuits
  • integer multiplication
  • pebble games
  • Karatsuba's method


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