Fast quantum modular exponentiation

Abstract

We present a detailed analysis of the impact on quantum modular exponentiation of architectural features and possible concurrent gate execution. Various arithmetic algorithms are evaluated for execution time, potential concurrency, and space trade-offs. We find that to exponentiate an $n\text{\ensuremath{-}}\text{bit}$ number, for storage space $100n$ (20 times the minimum $5n$), we can execute modular exponentiation 200--700 times faster than optimized versions of the basic algorithms, depending on architecture, for $n=128$. Addition on a neighbor-only architecture is limited to $O(n)$ time, whereas non-neighbor architectures can reach $O(\mathrm{log}\phantom{\rule{0.2em}{0ex}}n)$, demonstrating that physical characteristics of a computing device have an important impact on both real-world running time and asymptotic behavior. Our results will help guide experimental implementations of quantum algorithms and devices.