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arXiv:2607.19129

Single Link Removal Perturbation in Szegedy Quantum Walk: from Graph Completeness Testing to Integrity Monitoring

Sara Giordano, Miguel A. Martin-Delgado

quant-phSubmitted Jul 21, 2026• Announced Jul 22, 2026• 0 citations
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Abstract

We present a rigorous perturbative analysis of the Szegedy quantum walk search algorithm on the complete graph with marked nodes, when a specific anomaly is present in the graph. This is motivated by the problem of monitoring the integrity of dense trusted communication networks with a quantum-assisted procedure. The topology of these networks is modeled as a complete graph, and the anomaly of interest is the disappearance of a single communication link which represents the minimal and spectrally hardest structural defect to detect. Building on the graph-completeness testing algorithm framework, we quantify how the removal of a single unmarked-unmarked edge propagates through the relevant spectral quantities of the Szegedy quantum walk. Denoting by $n$ the total number of nodes of the graph and by $m$ the number of marked nodes, we prove that the perturbation to the transition matrix has spectral norm $Θ(1/n)$, and that the gap eigenvalue undergoes a strictly negative first-order shift for every $n$ and every number of marked nodes $m$, providing a formal proof of a conjecture from our completeness testing algorithm work; in the regime $m = Θ(n)$ relevant for the search algorithm, this shift has magnitude $Θ(1/n^2)$. The corresponding eigenphase shift satisfies $Δθ_\star = Θ(1/n^2)$ in the same regime. We establish that the rotation angle of the effective subspace under this perturbation is $O(1/n)$ for $m = Θ(n)$. Finally, we bound the change in success probability to $O(1/\sqrt{n})$ in this same regime of marked nodes, and show that this bound is dominated by the geometric misalignment of the effective subspace rather than by the spectral shift of the eigenphase. These results provide both the theoretical foundations and the fundamental scaling limits of quantum walk-based topology integrity monitoring under minimal structural perturbations.

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Quantum Searching on Markov Chains - The Complete Graph

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We will give an introduction into the quantum search algorithms on the Markov chains introduced by Szegedy and recent modifications based on partially absorbing Markov chains due to Krovi et al.Algorithmica 74, 851 (2016) It has been shown that a quantum search can find a set of marked vertices quadratically faster (in units of the hitting time) for any reversible Markov chain.The proofs are based on certain properties of the stationary state of the partially absorbing walk and rely on quantum phase estimation techniques.We will offer an alternative view of the underlying mechanism of the quantum search based on spectral properties of the quantum walk operator.By considering the complete graph as an example, we identify the relevant two-level quantum subspace leading to a Grover-like rotation in the underlying vector space.

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We present a rigorous perturbative analysis of the Szegedy quantum walk search algorithm on the complete graph with marked nodes, when a specific anomaly is present in the graph. This is motivated by the problem of monitoring the integrity of dense trusted communication networks with a quantum-assisted procedure. The topology of these networks is modeled as a complete graph, and the anomaly of interest is the disappearance of a single communication link which represents the minimal and spectrally hardest structural defect to detect. Building on the graph-completeness testing algorithm framework, we quantify how the removal of a single unmarked-unmarked edge propagates through the relevant spectral quantities of the Szegedy quantum walk. Denoting by nnn the total number of nodes of the graph and by mmm the number of marked nodes, we prove that the perturbation to the transition matrix has spectral norm Θ(1/n)Θ(1/n)Θ(1/n), and that the gap eigenvalue undergoes a strictly negative first-order shift for every nnn and every number of marked nodes mmm, providing a formal proof of a conjecture from our completeness testing algorithm work; in the regime m=Θ(n)m = Θ(n)m=Θ(n) relevant for the search algorithm, this shift has magnitude Θ(1/n2)Θ(1/n^2)Θ(1/n2). The corresponding eigenphase shift satisfies Δθ⋆=Θ(1/n2)Δθ_\star = Θ(1/n^2)Δθ⋆​=Θ(1/n2) in the same regime. We establish that the rotation angle of the effective subspace under this perturbation is O(1/n)O(1/n)O(1/n) for m=Θ(n)m = Θ(n)m=Θ(n). Finally, we bound the change in success probability to O(1/n)O(1/\sqrt{n})O(1/n​) in this same regime of marked nodes, and show that this bound is dominated by the geometric misalignment of the effective subspace rather than by the spectral shift of the eigenphase. These results provide both the theoretical foundations and the fundamental scaling limits of quantum walk-based topology integrity monitoring under minimal structural perturbations.
Testing graph completeness is a critical problem in computer science and network theory. Leveraging quantum computation, we present an efficient algorithm using the Szegedy quantum walk and quantum phase estimation (QPE). Our algorithm, which takes the number of nodes and the adjacency matrix as input, constructs a quantum walk operator and applies QPE to estimate its eigenvalues. These eigenvalues reveal the graph’s structural properties, enabling us to determine its completeness. We establish a relationship between the number of nodes in a complete graph and the number of marked nodes, optimizing the success probability and running time. The time complexity of our algorithm is O ( log 2 n ) , where n is the number of nodes of the graph. offering a clear quantum advantage over classical methods. This approach is useful in network structure analysis, evaluating classical routing algorithms, and assessing systems based on pairwise comparisons.
This paper studies the search for a single arc in a graph using the Szegedy walk. Arc search can be interpreted as finding a quantum particle not only in its position but also with a specific internal state. The quantum walk employed in this study is essentially the model proposed by Segawa and Yoshie for the purpose of edge search. First, we investigate how the symmetry of a graph is reflected in its time evolution matrix, and provide a sufficient condition under which the success probability of the search is independent of the marked arc. In particular, we prove that if a graph is arc-transitive, the success probability is independent of the choice of the marked arc. Next, we analyze path and cycle graphs and show that the quantum search is ineffective for these graphs, whereas it performs well for complete bipartite graphs Kn,nK_{n,n}Kn,n​. These results provide a theoretical foundation for studying arc and edge searches on various graphs, while also suggesting new problems concerning the eigenvalue analysis of edge-signed graphs in spectral graph theory.
© 2017, Editorial Department of Journal of Southeast University. All right reserved. Quantum computing thinking is used to explore new search methods for graph structures. A search algorithm for structural anomalies on complete graphs based on scattering quantum walk is proposed. Adding a pendant vertex to a complete graph, in which the number of vertices is N, breaks the symmetry of the complete graph and indicates the change of the topology of the complete graph. First, the precise definition of the evolutionary unitary operator U of scattering quantum walk in the complete graph is presented. The Hilbert space in which the walk occurs is projected to a lower-dimensional invariant subspace S, and the action of the evolutionary operator in the subspace US is given. Then, the equal superposition of all the basis states in the complete graph is chosen as the initial state. The eigenvalues and the eigenstates of US is calculated by using the perturbation theory, and the final state (pendants) of the algorithm is solved. Finally, the time complexity and the success probability of the algorithm are analyzed. The algorithm analysis and the Matlab simulation results show that the quantum search algorithm using scattering quantum walk can find the anomaly in O(N1/2) steps with the probability near to 1. However, the time complexity of the classical algorithm for searching the anomaly by the adjacency list is O(N). Therefore, the search algorithm based on scattering quantum walk can provide a quadric speed up over the classical algorithm for this specific problem.
Quantum walks have been proven to be a useful framework in designing new quantum algorithms, of which the search algorithm is the most notable. Besides a general search for a special vertex, recent researches have shown that quantum walks can also be used to find structural anomalies. Suppose a vertex of complete graph KN is attached to a second graph G, then the kind of structure anomaly will break the symmetry of the complete graph. The search algorithm based on scattering quantum walk model is presented to speed up locating this kind of structure anomaly. The concepts of scattering quantum walk model and collapsed graphs are presented. The definition of the evolutionary operator, which is different from that of a general search, is given. Based on the specific definition of evolutionary operator and the obvious symmetry of complete graph, it is shown that the search space is confined to a low-dimensional collapsed space, and the initial state is chosen to lie in this subspace. To illustrate the evolutionary process of the search algorithm, an example is given in the case that G is a single vertex. Taking advantage of our earlier study on the evolutionary operator of coined quantum walks with Grover coin, calculations of the unitary operator in the collapsed space are greatly simplified. To quantify the evolutionary process of the algorithm, we use the matrix perturbation theory involving a perturbative approach to find the eigenvalues and eigenstates. It is the degenerate zeroth-order eigenvalue 0 = 1 that leads to the interesting parts of the Hilbert space. Most of the recent researches of searching the structure anomalies focus on star graph SN with an unspecified graph G attached to one of its external vertices, where the overall graph is divided into two parts by the central vertex. It is shown that quantum speedup will occur if and only if the eigenvalues associated with these two parts in the N limit are the same. In this paper, we find that the collapsed graph of complete graphs can also be divided into two parts by a single collapsed vertex. As these two parts roughly correspond to the initial state and the desired state respectively, the techniques and results in star graphs can be generalized to the collapse graph on complete graph. What is more, under our definition of unitary evolution operator these two parts in the N limit will always share the same eigenvalue, i.e. 0 = 1, no matter what the structure of graph G is. Based on this, we prove that the search algorithm can find the target vertex in O(N) time steps with a success probability of 1-O(1N). That is to say, the quantum search algorithm gains a quadratic speedup over classical counterpart.
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.
We will give an introduction into the quantum search algorithms on the Markov chains introduced by Szegedy and recent modifications based on partially absorbing Markov chains due to Krovi et al.Algorithmica 74, 851 (2016) It has been shown that a quantum search can find a set of marked vertices quadratically faster (in units of the hitting time) for any reversible Markov chain.The proofs are based on certain properties of the stationary state of the partially absorbing walk and rely on quantum phase estimation techniques.We will offer an alternative view of the underlying mechanism of the quantum search based on spectral properties of the quantum walk operator.By considering the complete graph as an example, we identify the relevant two-level quantum subspace leading to a Grover-like rotation in the underlying vector space.