Abstract
Quantum walks are analogous to the classical random walks, and have important applications in quantum computation and quantum information. In this chapter, we give a gentle introduction to the first model of quantum walks—known as the coined model—, presented for the first time in 1993. Then, we consider the staggered model, which is much more recent and generalizes the most important types of quantum walks.
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Notes
- 1.
If we measure the quantum walker after each step, then we recover the probability distribution of the classical random walker.
- 2.
For the two-dimensional torus, we easily can adapt Eq. (5.13) by using modular arithmetics.
- 3.
A clique a graph Γ(V, E) is a subset of V such that any two vertices in the subset are adjacent, that is, a clique induces a complete subgraph.
- 4.
It is possible to strengthen the power of the coined model by using the square of the evolution operator [3].
- 5.
A line graph of a graph Γ (called root graph) is another graph L(Γ) so that each vertex of L(Γ) represents an edge of Γ and two vertices of L(Γ) are adjacent if and only if their corresponding edges share a common vertex in Γ.
- 6.
A clique graph K(G) of a graph G is a graph such that every vertex represents a maximal clique of G and two vertices of K(G) are adjacent if and only if the corresponding maximal cliques in G share at least one vertex in common.
- 7.
A k-colorable graph is the one whose vertices can be colored with at most k colors so that no two adjacent vertices share the same color.
- 8.
At this point, it is highly recommended some computer algebra system, such as Sage (for those that prefer free software) or Mathematica or Maple.
- 9.
In this context, Γ can be a multigraph.
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de Lima Marquezino, F., Portugal, R., Lavor, C. (2019). Quantum Walks. In: A Primer on Quantum Computing. SpringerBriefs in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-030-19066-8_5
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