Quantum Walk Sampling by Growing Seed Sets
Abstract
This work describes a new algorithm for creating a superposition over the edge set of a graph, encoding a quantum sample of the random walk stationary distribution. The algorithm requires a number of quantum walk steps scaling as $\widetilde{O}(m^{1/3} \delta^{1/3})$, with $m$ the number of edges and $\delta$ the random walk spectral gap. This improves on existing strategies by initially growing a classical seed set in the graph, from which a quantum walk is then run. The algorithm leads to a number of improvements: (i) it provides a new bound on the setup cost of quantum walk search algorithms, (ii) it yields a new algorithm for $st$connectivity, and (iii) it allows to create a superposition over the isomorphisms of an $n$node graph in time $\widetilde{O}(2^{n/3})$, surpassing the $\Omega(2^{n/2})$ barrier set by index erasure.
 Publication:

arXiv eprints
 Pub Date:
 April 2019
 arXiv:
 arXiv:1904.11446
 Bibcode:
 2019arXiv190411446A
 Keywords:

 Quantum Physics;
 Computer Science  Data Structures and Algorithms
 EPrint:
 14 pages