Quantum schoolbook multiplication with fewer Toffoli gates

TL;DR

This paper introduces a quantum circuit design for schoolbook multiplication that significantly reduces Toffoli gate count by using controlled add-subtract circuits, especially beneficial for small register sizes.

Contribution

It proposes a novel controlled add-subtract circuit that halves the Toffoli count compared to traditional methods, improving efficiency in quantum multiplication circuits.

Findings

Reduces Toffoli count by up to 30% for 256-bit key circuits

Uses controlled add-subtract circuits requiring only n-1 Toffoli gates

Advantageous for small register sizes despite better asymptotic algorithms

Abstract

This paper presents a method for constructing quantum circuits for schoolbook multiplication using controlled add-subtract circuits, asymptotically halving the Toffoli count compared to traditional controlled-adder-based constructions. Controlled n-qubit add-subtract circuits, which perform an addition when the control qubit is one and a subtraction when it is zero, require only n-1 Toffoli gates, instead of the 2n-1 needed for controlled adders. Despite the existence of multiplication circuits with better asymptotic scaling, schoolbook multiplication yields the lowest Toffoli counts for small register sizes, making it advantageous in practical applications. For example, the presented approach reduces the Toffoli count by up to around 30% in circuits for breaking 256-bit elliptic curve keys compared to circuits with standard schoolbook multipliers.