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arXiv:2410.22011 · Physical Review A

Complex-phase extensions of the Szegedy quantum walk on graphs

Sergio A. Ortega, Miguel Angel Martin-Delgado

Published Mar 19, 2025• 2 citations
arXivHTMLPDF

Abstract

This work introduces a graph-phased Szegedy's quantum walk, which incorporates link phases and local arbitrary phase rotations (APR), unlocking new possibilities for quantum algorithm efficiency. We demonstrate how to adapt quantum circuits to these advancements, allowing phase patterns that ensure computational practicality. The graph-phased model broadens the known equivalence between coined quantum walks and Szegedy's model, accommodating a wider array of coin operators. Through illustrative examples, we reveal intriguing disparities between classical and quantum interpretations of walk dynamics. Remarkably, local APR phases emerge as powerful tools for marking graph nodes, optimizing quantum searches without altering graph structure. We further explore the surprising nuances between single and double operator approaches, highlighting a greater range of compatible coins with the latter. To facilitate these advancements, we present an improved classical simulation algorithm, which operates with superior efficiency. This study not only refines quantum walk methodologies but also paves the way for future explorations, including potential applications in quantum search and PageRank algorithms. Our findings illuminate the path towards more versatile and powerful quantum computing paradigms.

DOI
10.1103/physreva.111.032216

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This work introduces a graph-phased Szegedy's quantum walk, which incorporates link phases and local arbitrary phase rotations (APR), unlocking new possibilities for quantum algorithm efficiency. We demonstrate how to adapt quantum circuits to these advancements, allowing phase patterns that ensure computational practicality. The graph-phased model broadens the known equivalence between coined quantum walks and Szegedy's model, accommodating a wider array of coin operators. Through illustrative examples, we reveal intriguing disparities between classical and quantum interpretations of walk dynamics. Remarkably, local APR phases emerge as powerful tools for marking graph nodes, optimizing quantum searches without altering graph structure. We further explore the surprising nuances between single and double operator approaches, highlighting a greater range of compatible coins with the latter. To facilitate these advancements, we present an improved classical simulation algorithm, which operates with superior efficiency. This study not only refines quantum walk methodologies but also paves the way for future explorations, including potential applications in quantum search and PageRank algorithms. Our findings illuminate the path towards more versatile and powerful quantum computing paradigms.
Quantum walks provide a versatile framework for quantum algorithms across a wide range of applications. We develop efficient classical simulation methods for Szegedy quantum walks that avoid explicit construction of the full unitary evolution operator. Unlike previous approaches restricted to a particular walk formulation, our framework is built from fundamental update and reflection operators, enabling the simulation of a broader class of Szegedy walk formulations. We further extend these methods to phase-estimation-based algorithms coupled to the walk, including implementations suitable for large sparse graphs. The resulting methods achieve optimal O(N2)O(N^2)O(N2) complexity for dense graphs with NNN nodes. For sparse graphs, the computational cost scales linearly with the number of edges, which is O(N)O(N)O(N) in many cases. We implement the framework in the Python package SQWLib and illustrate its capabilities through simulations of representative algorithms, including quantum simulated annealing and quantum search on graphs. These results provide a practical tool for studying Szegedy-walk-based algorithms numerically beyond purely analytical treatments.
The quantization of the PageRank algorithm is a promising tool for a future quantum internet. Here we present a modification of the quantum PageRank, introducing arbitrary phase rotations (APR) in the underlying Szegedy's quantum walk. We define three different APR schemes with only one phase as a degree of freedom. We have analyzed the behavior of these algorithms in a small generic graph, observing that a decrease of the phase reduces the standard deviation of the instantaneous PageRank, so the nodes of the network can be distinguished better. However, the algorithm takes more time to converge, so the phase cannot be decreased arbitrarily. With these results we choose a concrete value for the phase to later apply the algorithm to complex scale-free graphs. In these networks, the original quantum PageRank is able to break the degeneracy of the residual nodes and detect secondary hubs that the classical algorithm suppresses. Nevertheless, not all of the detected secondary hubs are real according to the PageRank's definition. Some APR schemes can overcome this problem, restoring the degeneration of the residual nodes and highlighting the truly secondary hubs of the networks. Finally, we have studied the stability of the new algorithms. The original quantum algorithm was known to be more stable than the classical. We have found that one of our algorithms, whose PageRank distribution resembles the classical one, has a stability similar to the original quantum algorithm.
Quantum mechanics empowers the emergence of quantum advantages in various fields, including quantum algorithms. Quantum PageRank is a promising tool for a future quantum internet. Recently, arbitrary phase rotations (APRs) have been introduced in the underlying Szegedy's quantum walk of the quantum PageRank algorithm. In this work we thoroughly study the role APR plays in quantum PageRank. We discover the versatility resulting from quantumness. Specifically, we discover the emergence of a cluster phenomenon in rankings considering the rotation phases, i.e., the existence of similar clusters in the distribution of the rankings and their fidelity with the corresponding classical PageRanks, the ranking distribution variance, the coherence and entanglement of PageRank states, and the power-law parameter in the ranking distributions on a scale-free network concerning the two rotation phases. Furthermore, we propose an alternate quantum PageRank with APR which provides an extra tunnel for the analysis of PageRank. We also study the PageRank on the trackback graph of a scale-free graph for the investigation of network information traffic tracking. We demonstrate the rich cluster diversity formed in our alternate quantum PageRank, which offers a novel perspective on the quantum versatility of PageRank. Our results present the quantum-enabled perspective for PageRanking and shed light on the design and application of practical quantum PageRank algorithms. Published by the American Physical Society 2024
Quantum walk search on graphs is primarily limited to real evolution operators, although there is no restriction for using the complex unitary matrix as the evolution operator. In the context of quantum search, several studies on the Grover’s search utilize the complex unitary matrix as an evolution operator, which allows for efficient searching of marked elements. However, the generalization of the quantum walk search to complex space has largely been overlooked. In this paper, we extend existing discrete-time coined quantum walk search algorithms on one- and two-dimensional periodic lattices to complex space by introducing two phase parameters, [Formula: see text], and [Formula: see text], into the coin and oracle operators, respectively, for four different discrete-time coined quantum walk search algorithms. One of the advantages of this generalization is that it enables the Grover and SKW coins with specific values of the phase parameters [Formula: see text], and [Formula: see text] to search for the exceptional configurations on a two-dimensional periodic lattice with high success probability, which is otherwise impossible to search using the real Grover and SKW coins operators.
This paper investigates the application of quantum walk algorithms to large-scale graph search problems. Classical algorithms often struggle with the computational complexity of exploring vast graphs, particularly when searching for specific patterns or nodes. Quantum walks, inspired by the probabilistic nature of random walks, provide a promising alternative by utilizing quantum superposition and interference to accelerate the search process. We explore the theoretical framework and potential advantages of employing quantum walks for graph traversal, focusing on scenarios where exponential speedups are theoretically possible. This work aims to demonstrate the potential of quantum computing to tackle computationally demanding graph search tasks that are currently intractable for classical methods. The core concept leverages the ability to simultaneously explore multiple paths within a graph, drastically reducing the search time compared to sequential exploration. We present a foundational understanding of quantum walk implementation and analyze its suitability for various graph search algorithms.
This paper explores the application of quantum walk algorithms to graph traversal and search problems. Classical algorithms often struggle with the computational complexity of traversing and searching large graphs, leading to significant time and resource limitations. Quantum walks, utilizing superposition and interference, offer a potential pathway to achieve exponential speedups. This work details the theoretical framework for applying quantum walks to graph problems, specifically focusing on traversal and search tasks. We present a generalized approach for mapping graph structures onto quantum walk systems and discuss the key mathematical formulations involved. The core of our investigation revolves around the construction of quantum walk operators that mimic the movement and interaction rules inherent in graph traversal. We analyze the evolution of the quantum walk state and derive expressions for the probability of reaching a specific node in the graph. Furthermore, we examine the impact of graph topology on the performance of the quantum walk, highlighting the potential for tailored algorithms to exploit specific graph structures. The results demonstrate the theoretical feasibility of quantum walk algorithms for significantly accelerating graph traversal and search operations, contributing to advancements in areas such as network analysis, database management, and artificial intelligence.