Constant-optimized quantum circuits for modular multiplication and exponentiation

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 Θ(n2). 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.