Factoring using 2n+2 qubits with Toffoli based modular multiplication

TL;DR

This paper presents a space-efficient implementation of Shor's quantum factoring algorithm using only 2n+2 qubits with a purely Toffoli-based modular multiplication circuit, enabling easier testing and fault localization.

Contribution

It introduces a novel in-place constant-adder using dirty ancilla qubits and achieves modular multiplication with purely Toffoli gates, reducing overheads in quantum factoring.

Findings

Achieves O(n^3) circuit depth and O(n^3 log(n)) gate count.

Uses only 2n+2 qubits, matching previous space-optimized methods.

Enables fault testing at both logical and hardware levels.

Abstract

We describe an implementation of Shor's quantum algorithm to factor n-bit integers using only 2n+2 qubits. In contrast to previous space-optimized implementations, ours features a purely Toffoli based modular multiplication circuit. The circuit depth and the overall gate count are in O(n^3) and O(n^3 log(n)), respectively. We thus achieve the same space and time costs as Takahashi et al., while using a purely classical modular multiplication circuit. As a consequence, our approach evades most of the cost overheads originating from rotation synthesis and enables testing and localization of faults in both, the logical level circuit and an actual quantum hardware implementation. Our new (in-place) constant-adder, which is used to construct the modular multiplication circuit, uses only dirty ancilla qubits and features a circuit size and depth in O(n log(n)) and O(n), respectively.