Quantum community detection via deterministic elimination

TL;DR

The paper addresses the challenge of detecting community structure and botnets in large complex networks. It introduces deteQt, a quantum protocol that starts from a modularity-based graph representation, prepares the leading eigenvector as a ground state, signs its amplitudes via quantum signal processing to yield a signed real-weighted state, and then deterministically reads out the partition using a hypergraph-state LCU readout. Key contributions include a complete subroutine chain (input encoding, ground-state preparation, signing, and two readout strategies), a scalable scaling analysis for qubits and gates, and demonstration of botnet detection in sizable networks. The approach provides a building block for quantum-accelerated graph analysis with potential polynomial-sample advantages in detecting anomalous subgraphs, contributing to cybersecurity applications in large networks.

Abstract

We propose a quantum algorithm for calculating the structural properties of complex networks and graphs. The corresponding protocol -- deteQt -- is designed to perform large-scale community and botnet detection, where a specific subgraph of a larger graph is identified based on its properties. We construct a workflow relying on ground state preparation of the network modularity matrix or graph Laplacian. The corresponding maximum modularity vector is encoded into a $\log(N)$-qubit register that contains community information. We develop a strategy for ``signing'' this vector via quantum signal processing, such that it closely resembles a hypergraph state, and project it onto a suitable linear combination of such states to detect botnets. As part of the workflow, and of potential independent interest, we present a readout technique that allows filtering out the incorrect solutions deterministically. This can reduce the scaling for the number of samples from exponential to polynomial. The approach serves as a building block for graph analysis with quantum speed up and enables the cybersecurity of large-scale networks.

Figures (8)

• Figure 1: Flowchart summarizing deteQt --- the developed quantum algorithms for modularity-based community and botnet detection. The protocol consists of several steps, including finding the leading eigenvector of the modularity matrix, and readout sampling techniques to infer its signs. We detail each step and subroutine in Sec. \ref{['sec:protocol']}.
• Figure 2: (a) Full circuit for projecting the signed max vector onto the hypergraph superposition state. The first layer of Hadamards prepares the state $|+\rangle^{\otimes n}$, which the hypergraph circuits act upon [as in Eq. \ref{['hypergraph_eq']}]. (b) Circuit $\hat{U}_\lambda$ for the projection of the signed max vector onto the system register (when $\boldsymbol{0}$ is measured on the $(n+4)$ ancilla qubits). Here $\hat{P}_{0} = \ket{0}^{\otimes 2n+3}\bra{0}^{\otimes 2n+3}$ and $H \equiv H^{\otimes n}$. (c) LCU circuit for the preparation of the hypergraph superposition state. The LCU$_h$ block has the standard form of the LCU circuit but without the final PREPARE operation (in this case, a layer of Hadamards). The top register is then sampled to retrieve the bipartition. (d) Mapping between the hypergraphs and the circuits preparing the hypergraph states which make up the LCU. Black lines represent edges between the nodes of the hypergraph. These translate into CZ gates (black connected dots) between corresponding qubits in the hypergraph state generation. Shaded regions correspond to hyperedges between multiple qubits, e.g. 5-6-7 and 3-4-5-6 in the figure. These translate into multi-controlled phase gates. The multi-qubit CZ gates act on a $|+\rangle^{\otimes 7}$ state prepared from the computational zero state with Hadamards.
• Figure 3: Comparison of the zero-overlap selection and small-overlap section methods, for $N=16$ nodes and botnet size $k=3$. We have $\binom{16}{3}=560$ candidate states, and we visualize the overlaps of these states with the states in the hypergraph LCU. For both methods, the black rectangles appear at different positions on each row, indicating that each candidate state has a unique set of zeros/small overlaps $X_2^{G_c}$. In the zero-overlap case, we select hypergraph states with botnet size $k_{\mathrm{LCU}} = \frac{16}{2}-3 = 5$, giving an LCU with $\binom{16}{5}=4368$ unitaries. In the small-overlap case, we select hypergraph states with botnet size $k_{\mathrm{LCU}} = 1$, giving an LCU with $\binom{16}{1}=16$ unitaries. Changing from zero-overlap to small-overlap selection gives us an exponential decrease in the LCU depth, which is offset by an increase in the number of circuit runs.
• Figure 4: Example of a deteQt run using small-overlap selection. The network considered consists of $N=10$ nodes (vertices), 18 interactions (edges), and a hidden botnet of size $k=3$. We run deteQt 20 times (trials) reading out the estimated partitions A and B. By counting the occurrences of the single nodes in the partitions, we build the probability distribution to estimate the botnet composition (in our example, 2-5-8).
• Figure 5: Example of a deteQt run to detect botnets of different types and sizes. We show botnets that mix with the network and behave similarly to healthy nodes (hidden botnets) or botnets whose intranet is isolated from the rest of the network, limiting their exposure to the healthy nodes (isolated botnets). In the case of the hidden botnet ($k=5$), the network studied is composed of $N=50$ nodes (vertices) and 1213 interactions (edges) with a total of more than two million possible partitions. In the case of the isolated botnet ($k=4$), the network studied is composed of $N=100$ nodes and 4624 interactions, with a total of approximately four million possible partitions.