Generator of an abstract quantum walk
Etsuo Segawa, Akito Suzuki

TL;DR
This paper introduces a framework for abstract quantum walks using a unitary operator that defines a directed graph and analyzes the walk's asymptotic behavior through its generator, generalizing Szegedy's model.
Contribution
It provides a method to derive the generator of an abstract quantum walk, extending the Szegedy evolution operator to a broader class of quantum walks.
Findings
Defines a directed graph from the unitary operator of a quantum walk.
Derives the generator of the quantum walk's evolution, linking it to asymptotic behavior.
Generalizes Szegedy's evolution operator to a new class of quantum walks.
Abstract
We consider an abstract quantum walk defined by a unitary evolution operator , which acts on a Hilbert space decomposed into a direct sum of Hilbert spaces . We show that such naturally defines a directed graph and the probability of finding a quantum walker on . The asymptotic property of an abstract quantum walker is governed by the generator of such that . We derive the generator of an evolution of the form , a generalization of the Szegedy evolution operator. Here is a boundary operator and a shift operator.
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Taxonomy
TopicsQuantum Computing Algorithms and Architecture · Quantum Information and Cryptography · Quantum many-body systems
