Qlustering: Harnessing Network-Based Quantum Transport for Data Clustering

TL;DR

Qlustering addresses unsupervised clustering by leveraging quantum transport in a network: data are encoded as input states in a tight-binding Hamiltonian $\mathcal{H}$ and propagated under the Lindblad equation $\dot\rho = -i[\mathcal{H},\rho] + \mathcal{L}[\rho]$, with cluster identities read from steady-state currents $J[\Psi_n]$. The method iteratively perturbs $\mathcal{H}$ to minimize a current-based cost $CF(J,N)$, potentially using multiple particles and a consensus scheme for stability. The authors validate on synthetic, localization, QM9, and Iris datasets, showing competitive or superior performance to $k$-means, particularly for non-convex or high-dimensional data, and highlight robustness, low complexity, and compatibility with photonic hardware as practical advantages. They also introduce a consensus clustering step to enhance stability and provide public code for replication.

Abstract

We introduce Qlustering, a quantum-inspired algorithm for unsupervised learning that leverages network-based quantum transport to perform data clustering. In contrast to traditional distance-based methods, Qlustering treats the steady-state dynamics of quantum particles propagating through a network as a computational resource. Data are encoded as input states in a tight-binding Hamiltonian framework governed by the Lindblad master equation, and cluster assignments emerge from steady-state output currents at terminal nodes. The algorithm iteratively optimizes the network's Hamiltonian to minimize a physically motivated cost function, achieving convergence through stochastic updates. We benchmark Qlustering on synthetic datasets, a localization problem, and real-world chemical and biological data, namely subsets of the QM9 molecular database and the Iris dataset. Across these diverse tasks, Qlustering demonstrates competitive or superior performance compared with classical methods such as k-means, particularly for non-convex or high-dimensional data. Its intrinsic robustness, low computational complexity, and compatibility with photonic implementations suggest a promising route toward physically realizable, quantum-native clustering architectures.

Figures (10)

• Figure 1: Schematic representation of the Qlustering network. An input state vector $\Psi$ of dimension $L$ is injected into the network through the input nodes. In this figure, $L=3$. The propagation, injection, and extraction of a particle are modeled by Eq. \ref{['eqn1']}. Here, $L$ denotes the number of input nodes and also the dimensionality of the state vectors, $q$ is the number of output nodes (i.e., clusters), and $M$ is the number of hidden nodes. After reaching a steady state, the current from each output node is computed. The state is then assigned to the group corresponding to the output node with the highest current.
• Figure 2: Frames captured from the Qlustering process applied to 60 state vectors in a 3-dimensional Hilbert space, grouped into 5 distinct clusters. Upper panels illustrate the spatial distribution of the vectors in the space, while the bottom panels illustrate the Qlustering network with its 3-2-5 node structure. The fitting process runs over iterations $t = 1, \dots, T$, where $T$ is a predefined number of steps. Panel (a) shows the initial state at $t = 1$, where 3 clusters are already correctly grouped. The corresponding Hamiltonian for this initial step is displayed in bottom panel of (a). In each subsequent iteration, a random entry in the Hamiltonian $\mathcal{H}$ is selected and modified. If the new Hamiltonian yields a lower cost function $CF$ (Eq. \ref{['CF']}), it is retained and used in the next iteration. Bottom panel (b) presents the Hamiltonian at $t = 9$, with the two changes from the initial Hamiltonian highlighted in yellow. The resulting clustering at $t = 9$ is shown in upper panel (b). The panels of (c) illustrate the clustering and Hamiltonian at iteration $t = 12$.
• Figure 3: Qlustering of 3-dimensional vectors into four groups at $\omega = 0.15$. (a) Spatial distribution of the input state vectors. (b) Consensus matrix from 10 repeated Qlustering runs. Yellow regions indicate high co-clustering frequency, while blue denotes low agreement. Each square represents a vector pair, with color intensity indicating the frequency of co-clustering (yellow: high consistency; blue: low agreement). A stable Qlustering pattern is observed, with four distinct block structures emerge in the consensus matrix, reflecting strong consistency in group assignments across runs. Qlustering of 2-dimensional vectors into three groups at $\omega = 0.2$.
• Figure 4: Top: Qlustering scores across varying group widths, $\omega$. Solid-line triangles indicate the consensus clustering scores for the Rand Index (RI) and Adjusted Rand Index (ARI), while dashed-line circles represent the corresponding mean values. High clustering performance is maintained until significant overlap between groups occurs (marked by the transparent red line), after which scores decline toward random clustering levels as the distribution approaches uniformity. Bottom: The overlap point visualized. Spatial distribution of five groups in a three-dimensional Hilbert space at $\omega = 0.25$, marking the onset of the overlap point. Colors distinguish the clusters. The groups remain largely separated---i.e., no true overlap yet---though some points already lie at approximately equal distances from their own center and that of another group. This regime marks the transition where Qlustering performance begins to decline from perfect scores toward reduced accuracy, as shown in the top figure.
• Figure 5: Qlustering performance on the localization task. Rand Index (RI) is shown in blue and Adjusted Rand Index (ARI) in orange on varying $\Delta_{IPR}$. The network clusters 10-dimensional wavefunctions with IPR values ranging from 1 to 10. As $\Delta_{IPR}$ decreases, performance declines. This drop corresponds to the mixing of strongly localized and delocalized states, which reduces the distinctiveness of the underlying physical classes.