[Paper Review] Quantum Walks on Embeddings

Hanmeng Zhan|arXiv (Cornell University)|Nov 23, 2017

Quantum Computing Algorithms and Architecture27 references3 citations

TL;DR

This paper introduces vertex-face walks, a novel discrete quantum walk model defined on orientable graph embeddings using arc-face and arc-tail incidence structures. By constructing a unitary transition matrix via reflections on normalized incidence matrices, the authors establish a spectral correspondence and show that the principal logarithm of $U^2$ yields an oriented graph when the vertex-face structure forms a partial geometric design, revealing connections to quantum search and non-classical dynamics.

ABSTRACT

We introduce a new type of discrete quantum walks, called vertex-face walks, based on orientable embeddings. We first establish a spectral correspondence between the transition matrix $U$ and the vertex-face incidence structure. Using the incidence graph, we derive a formula for the principal logarithm of $U^2$, and find conditions for its underlying digraph to be an oriented graph. In particular, we show this happens if the vertex-face incidence structure forms a partial geometric design. We also explore properties of vertex-face walks on the covers of a graph. Finally, we study a non-classical behavior of vertex-face walks.

Motivation & Objective

• To define a new discrete quantum walk model—vertex-face walks—based on orientable graph embeddings.
• To establish a spectral correspondence between the transition matrix $U$ and the vertex-face incidence structure.
• To investigate conditions under which the underlying digraph of $\log(U^2)$ is an oriented graph, particularly linking this to partial geometric designs.
• To explore the dynamics of vertex-face walks on graph covers and compare them to classical random walks.
• To extend the model to non-orientable embeddings using graph-encoded maps (gems) and analyze its reducibility and equivalence to walks on flag graphs.

Proposed method

• Define the arc-face incidence matrix $M$ and arc-tail incidence matrix $N$ from a consistent orientation of face boundaries in an orientable embedding.
• Normalize $M$ and $N$ to obtain $\widehat{M}$ and $\widehat{N}$, then construct the unitary transition matrix $U = (2\widehat{M}\widehat{M}^T - I)(2\widehat{N}\widehat{N}^T - I)$.
• Use the incidence graph of the vertex-face structure to derive a formula for the principal logarithm of $U^2$.
• Characterize when the digraph of $\log(U^2)$ is oriented, showing this occurs if the vertex-face structure forms a partial geometric design.
• Analyze vertex-face walks on graph covers using spectral and combinatorial techniques.
• Extend the model to non-orientable embeddings via graph-encoded maps (gems), where flags represent vertices and the walk is defined on a 3-edge-colored cubic graph.

Experimental results

Research questions

• RQ1 Under what conditions is the digraph of $\log(U^2)$ an oriented graph?
• RQ2 How does the spectral structure of the vertex-face walk relate to the combinatorial properties of the embedding?
• RQ3 What is the relationship between vertex-face walks and continuous quantum walks, particularly when $U = \exp(tS)$ for a skew-adjacency matrix $S$?
• RQ4 How do vertex-face walks compare to classical random walks in terms of dynamics and search efficiency?
• RQ5 Can the vertex-face walk model be generalized to non-orientable embeddings, and what structural properties emerge in such cases?

Key findings

• The transition matrix $U$ of the vertex-face walk is unitary and constructed as a product of two reflections: one onto the column space of the normalized arc-face incidence matrix and one onto the column space of the normalized arc-tail incidence matrix.
• A spectral correspondence is established between $U$ and the vertex-face incidence structure, enabling the derivation of a formula for the principal logarithm of $U^2$.
• The underlying digraph of $\log(U^2)$ is an oriented graph if and only if the vertex-face incidence structure forms a partial geometric design.
• Vertex-face walks exhibit non-classical behavior: they can remain localized near the initial state, in contrast to classical random walks that typically spread out.
• On the toroidal grid $C_n \square C_n$, vertex-face walks achieve higher success probabilities in search algorithms than arc-reversal walks, due to the ability to transition between non-adjacent arcs in a single step.
• For non-orientable embeddings, the vertex-face walk is equivalent to a quantum walk on the vertices of one component of the distance-2 graph of the gem (graph-encoded map), and the walk is reducible if and only if the original embedding is orientable.