High Performance Quantum Modular Multipliers

TL;DR

This paper introduces new quantum modular multipliers based on classical techniques, achieving exact results with efficient resource complexity, and provides an empirical analysis of their implementation costs for large inputs.

Contribution

It presents a novel set of reversible quantum modular multipliers derived from classical methods, with detailed resource analysis for large input sizes.

Findings

Exact modular multiplication for all binary inputs

Resource complexity comparable to non-modular multipliers

Empirical gate count and circuit depth analysis for 2048-bit inputs

Abstract

We present a novel set of reversible modular multipliers applicable to quantum computing, derived from three classical techniques: 1) traditional integer division, 2) Montgomery residue arithmetic, and 3) Barrett reduction. Each multiplier computes an exact result for all binary input values, while maintaining the asymptotic resource complexity of a single (non-modular) integer multiplier. We additionally conduct an empirical resource analysis of our designs in order to determine the total gate count and circuit depth of each fully constructed circuit, with inputs as large as 2048 bits. Our comparative analysis considers both circuit implementations which allow for arbitrary (controlled) rotation gates, as well as those restricted to a typical fault-tolerant gate set.