Improved reversible and quantum circuits for Karatsuba-based integer multiplication

Alex Parent, Martin Roetteler, Michele Mosca · 2017 · TQC 2017

| 编号 QM002 | 主题 模运算与乘法 | 链接 arXiv:1706.03419 |

摘要

摘要保留原文（英文）

摘要取自 arXiv / 出版方元数据，未作翻译，以保证与原文一致。

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.

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