Southwest Quantum Information and Technology

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

Presenting Author: Martin Roetteler, Microsoft
Contributing Author(s): Thomas Haener and Krysta M. Svore

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. We implemented and simulated a Toffoli network for the entire controlled modular multiplication piece of Shor's algorithm in LIQUi|>, for real-world bit-sizes of up to 8,192. Asymptotically, 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. Our resource estimates determine also the constants for the scaling of the circuit size.

Read this article online: https://arxiv.org/abs/1611.07995

(Session 9c : Friday from 5:45pm - 6:15pm)