Faster Quantum Concentration via Grover's Search

TL;DR

This paper introduces quantum algorithms leveraging Grover's search to significantly speed up routing concentration assignments in fat-and-slim concentrators, outperforming classical methods when capacity scales appropriately.

Contribution

The paper develops quantum algorithms for routing concentration assignments that are asymptotically faster than classical algorithms using Grover's search.

Findings

Quantum algorithms achieve $O(rac{1}{2}(1+ ta))$ time complexity.

Quantum algorithms outperform classical when $c  c  o(n)$.

Speedup depends on the capacity scaling parameter ta.

Abstract

We present quantum algorithms for routing concentration assignments on full capacity fat-and-slim concentrators, bounded fat-and-slim concentrators, and regular fat-and-slim concentrators. Classically, the concentration assignment takes $O(n)$ time on all these concentrators, where $n$ is the number of inputs. Powered by Grover's quantum search algorithm, our algorithms take $O(\sqrt{nc}\ln{c})$ time, where $c$ is the capacity of the concentrator. Thus, our quantum algorithms are asymptotically faster than their classical counterparts, when $c\ln^2{c}=o(n)$.In general, $c = n^\mu,$ satisfies $c\ln^2{c}=o(n),$ implying a time complexity of $O(n^{0.5(1+ \mu )} \ln n),$ for any $\mu, 0 < \mu < 1.$