Constant-Optimized Quantum Circuits for Modular Multiplication and Exponentiation

TL;DR

This paper presents optimized quantum circuits for modular multiplication and exponentiation, focusing on specific values of C and M to reduce circuit size and improve performance in quantum factoring algorithms.

Contribution

The authors develop bottom-up optimization techniques for modular multiplication circuits, achieving asymptotic size improvements and practical enhancements for quantum factoring implementations.

Findings

Some circuits are asymptotically smaller than previous ones

Constant-factor improvements in modular exponentiation circuits

Significant size reduction for small circuits in experimental settings

Abstract

Reversible circuits for modular multiplication $Cx$%$M$ with $x<M$ arise as components of modular exponentiation in Shor's quantum number-factoring algorithm. However, existing generic constructions focus on asymptotic gate count and circuit depth rather than actual values, producing fairly large circuits not optimized for specific $C$ and $M$ values. In this work, we develop such optimizations in a bottom-up fashion, starting with most convenient $C$ values. When zero-initialized ancilla registers are available, we reduce the search for compact circuits to a shortest-path problem. Some of our modular-multiplication circuits are asymptotically smaller than previous constructions, but worst-case bounds and average sizes remain $\Theta(n^2)$. In the context of modular exponentiation, we offer several constant-factor improvements, as well as an improvement by a constant additive term that is significant for few-qubit circuits arising in ongoing laboratory experiments with Shor's algorithm.