Accelerating Spectral Clustering on Quantum and Analog Platforms

TL;DR

Spectral clustering traditionally suffers from $O(N^3)$ eigendecomposition costs for large graphs. The authors propose a first-of-its-kind hybrid quantum-analog framework that performs wave dynamics on quantum/analog hardware and uses Dynamic Mode Decomposition on analog devices to recover Laplacian eigenpairs in $O(N)$ time. The pipeline consists of (1) evolving Schrödinger dynamics tied to the symmetric graph Laplacian, (2) extracting eigenvalues/eigenvectors via a reduced SVD and DMD, and (3) solving a linear system to obtain clustering coefficients, with quantum components offering potential polylogarithmic scaling in certain steps. The approach is demonstrated on benchmark networks, achieving close to ground-truth clustering with high accuracy, and is supported by a reproducible codebase. This work highlights the potential of cross-platform computing to tackle large-scale graph problems by exploiting the strengths of quantum and analog hardware in a unified algorithmic framework.

Abstract

We introduce a novel hybrid quantum-analog algorithm to perform graph clustering that exploits connections between the evolution of dynamical systems on graphs and the underlying graph spectra. This approach constitutes a new class of algorithms that combine emerging quantum and analog platforms to accelerate computations. Our hybrid algorithm is equivalent to spectral clustering and significantly reduces the computational complexity from $O(N^3)$ to $O(N)$, where $N$ is the number of nodes in the graph. We achieve this speedup by circumventing the need for explicit eigendecomposition of the normalized graph Laplacian matrix, which dominates the classical complexity, and instead leveraging quantum evolution of the Schrödinger equation followed by efficient analog computation for the dynamic mode decomposition (DMD) step. Specifically, while classical spectral clustering requires $O(N^3)$ operations to perform eigendecomposition, our method exploits the natural quantum evolution of states according to the graph Laplacian Hamiltonian in linear time, combined with the linear scaling for DMD that leverages efficient matrix-vector multiplications on analog hardware. We prove and demonstrate that this hybrid approach can extract the eigenvalues and scaled eigenvectors of the normalized graph Laplacian by evolving Schrödinger dynamics on quantum computers followed by DMD computations on analog devices, providing a significant computational advantage for large-scale graph clustering problems. Our demonstrations can be reproduced using our code that has been released at https://github.com/XingziXu/quantum-analog-clustering.

Figures (6)

• Figure 1: Schematic of AMC circuits for matrix-vector multiplication between matrix $\mathbf{A}$ and vector $\mathbf{x}$, where $\mathbf{A}=\mathbf{A}_+-\mathbf{A}_-$sun2022invited. $\mathbf{A}\in\mathcal{R}^{M\times N}$, $\mathbf{x}\in\mathcal{R}^{N\times 1}$, $\mathbf{y}\in\mathcal{R}^{M\times 1}$.
• Figure 2: The estimated and the ground truth eigenvector of the adjacency matrix of the Karate club network. Eigenvector (Node index)
• Figure 3: The wave dynamics of the second node on the Twitter dataset simulated with Qiskit dynamics verify the stability of quantum-simulated dynamics.
• Figure 4: Clusters of the Twitter interaction network for the US Congress, comparing the ground truth and the proposed method's results.
• Figure 5: Comparison of the ground truth spectral clustering results and from the proposed quantum-analog framework on $200$ nodes from the Facebook social network graph.

Theorems & Definitions (2)

• proof
• proof