Quantum Algorithm for Testing Graph Completeness

TL;DR

The paper addresses the problem of testing graph completeness using a quantum approach based on Szegedy's discrete-time quantum walk and Quantum Phase Estimation (QPE). It establishes a linear relation between the total number of nodes $n$ and the number of marked nodes $m^*$, ensuring that the hitting time aligns with the maximum marking probability and yielding a constant-time first stage. The proposed two-stage algorithm leverages Szegedy's walk with marked vertices, followed by a QPE verification that compares estimated eigenphases to the complete-graph benchmark, achieving an overall complexity of $O(\log^2 n)$. This polylogarithmic quantum advantage over classical $O(n^2)$ checks has practical implications for network analysis, graph traversal, clustering, and fairness in pairwise comparisons.

Abstract

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 $\mathcal{O}(\log^2n)$, 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.

Figures (7)

• Figure 1: Examples of complete graphs $K_{n}$ for low number of nodes $n =3,4,5,6$.
• Figure 2: (a) Original unmarked graph $\mathcal{G}$ and its duplicated graph for the Szegedy quantum walk. To each edge $(i,j)$ of the underlying graph correspond two edges of the bipartite one $(x_i, y_j)$ and $(y_i, x_j)$. (b) Similarly with a marked node in red. The marked vertices are absorbing states $s_i$ with only incoming directed links and no outgoing ones. In the duplicated graph the edges connecting the twin marked edges are added.
• Figure 3: The probability of finding the walker on a marked node reaches its maximum value at time $t^*_{\text{max}}$. Introducing the linear relationship presented in Eq.(\ref{['eq_linear_relationship_numerical']}) confines this maximum probability value, denoted as $P^*_{M}(t^*_{\text{max}},n)$, between the numerical limits $\lim_{n\to 1}P^*_{M}(t^*_{\text{max}},n)$ and $\lim_{n\to \infty}P^*_{M}(t^*_{\text{max}},n)$. In the plotted graph, the trend of $P^*_{M}(t^*_{\text{max}},n)$ is illustrated for $n$ values ranging from $1$ to $200$, along with its limits as $n$ approaches $1$ and $\infty$. The numerical values for these boundaries are provided in Eqs.(\ref{['eq_probability_boundaries_numerical_1']}) and (\ref{['eq_probability_boundaries_numerical_2']}), respectively.
• Figure 4: Flowchart of the completeness testing quantum algorithm as described in Subsection \ref{['sub_III_A']}. Input: $\left(n, P, m=\frac{n}{a}-1,t^*,\ket{\psi(0)},X\right)$, the number of nodes $n$ of the graph $\mathcal{G}$, the transition matrix $P$ of the graph $\mathcal{G}$, the linear relationship $m^*=\left[\frac{n}{a}-1\right]$ (Eq.(\ref{['eq_linear_relationship_numerical']})), the running time $t^*$, the initial quantum state for the quantum walk $\ket{\psi(0)}$ and the position operator. Output: A truth value, $\mathcal{C}$, concerning the completeness of the graph $\mathcal{G}$.
• Figure 5: Comparison between the number of bits $f$ and $l$ that we need to evaluate in the QPE. The number of bits $f$ is calculated from $\mathcal{F}(n)$, while the number of bits $l$ is calculated from the simulated data.

Theorems & Definitions (6)

• Definition 1
• Definition 2
• Definition 3
• Proposition 1
• Proposition 2
• Proposition 3