High Performance Quantum Modular Multipliers

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 [1], and 3) Barrett reduction [2]. 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.