Constant-Optimized 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 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.
- Publication:
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arXiv e-prints
- Pub Date:
- February 2012
- DOI:
- arXiv:
- arXiv:1202.6614
- Bibcode:
- 2012arXiv1202.6614M
- Keywords:
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- Computer Science - Emerging Technologies;
- Quantum Physics
- E-Print:
- 29 pages, 9 tables, 19 figures. Minor change: fixed two typos in the abstract and body