# Improved reversible and quantum circuits for Karatsuba-based integer multiplication

## TL;DR

This paper introduces a more space-efficient reversible quantum circuit for integer multiplication based on Karatsuba's method, reducing qubit requirements significantly while maintaining manageable operational overhead.

## Contribution

It presents a novel reversible circuit for integer multiplication that asymptotically reduces space complexity from O(n^{1.585}) to O(n^{1.427}) using Karatsuba's recursive approach.

## Key findings

- Reduces qubit space complexity in quantum integer multiplication circuits.
- Maintains low increase in operation count and depth.
- Achieves asymptotic improvements through pebble game analysis on ternary trees.

## Abstract

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.