ConstantOptimized Quantum Circuits for Modular Multiplication and Exponentiation
Abstract
Reversible circuits for modular multiplication $Cx$%$M$ with $x<M$ arise as components of modular exponentiation in Shor's quantum numberfactoring 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 bottomup fashion, starting with most convenient $C$ values. When zeroinitialized ancilla registers are available, we reduce the search for compact circuits to a shortestpath problem. Some of our modularmultiplication circuits are asymptotically smaller than previous constructions, but worstcase bounds and average sizes remain $\Theta(n^2)$. In the context of modular exponentiation, we offer several constantfactor improvements, as well as an improvement by a constant additive term that is significant for fewqubit circuits arising in ongoing laboratory experiments with Shor's algorithm.
 Publication:

arXiv eprints
 Pub Date:
 February 2012
 arXiv:
 arXiv:1202.6614
 Bibcode:
 2012arXiv1202.6614M
 Keywords:

 Computer Science  Emerging Technologies;
 Quantum Physics
 EPrint:
 29 pages, 9 tables, 19 figures. Minor change: fixed two typos in the abstract and body