Matrix product state has become the algorithm of choice when studying one-dimensional interacting quantum many-body systems, which demonstrates to be able to explore the most relevant portion of the exponentially large quantum Hilbert space and find accurate solutions. Here we propose a quantum inspired K-means clustering algorithm which first maps the classical data into quantum states represented as matrix product states, and then minimize the loss function using the variational matrix product states method in the enlarged space. We demonstrate the performance of this algorithm by applying it to several commonly used machine learning datasets and show that this algorithm could reach higher prediction accuracies and that it is less likely to be trapped in local minima compared to the classical K-means algorithm.
The method of choice to study one-dimensional strongly interacting many-body quantum systems is based on matrix product states and operators. Such a method allows one to explore the most relevant and numerically manageable portion of an exponentially large space. It also allows one to accurately describe correlations between distant parts of a system, an important ingredient to account for the context in machine learning tasks. Here we introduce a machine learning model in which matrix product operators are trained to implement sequence-to-sequence prediction, i.e., given a sequence at a time step, it allows one to predict the next sequence. We then apply our algorithm to cellular automata (for which we show exact analytical solutions in terms of matrix product operators) and to nonlinear coupled maps. We show advantages of the proposed algorithm when compared to conditional random fields and a bidirectional, long, short-term-memory neural network. To highlight the flexibility of the algorithm, we also show that it can readily perform classification tasks.
Spectral clustering is one of the most robust methods of an unsupervised machine learning field. Compared with the well-known k-means, spectral clustering uses the spectral properties of the Laplacian matrix to project data to a low-dimensional clustering with higher efficiency. However, for a sparse network with a substantial degree of heterogeneity, standard spectral clustering often overfits noise in the periphery of a sparse and stochastic graph. In our work, we combined the quantum version of the k-means algorithm with regularized spectral clustering and proposed a new quantum machine learning algorithm. It is more like a combination of classical and quantum algorithms. Compared with the established quantum k-means algorithm, our method has obvious computational advantages in non-convex or nested structures. At this stage, the research of quantum algorithms mainly stays in the simulation of classical computers. Our method does not need to expend a lot of effort to preprocess the data at the initial quantum state stage. Under the assumption of quantum data encoding, we theoretically demonstrate the feasibility of the quantum regularized spectral clustering algorithm. Then combine Q-means algorithm complexity advantages to complete the clustering task. This work aims to pave the path and help realize quantum spectral clustering algorithms on real quantum computers in the future and help create other versions of quantum machine learning algorithms. We provide simulations and data examples to illustrate these results.
Spectral clustering is a powerful unsupervised machine learning algorithm for clustering data with nonconvex or nested structures [A. Y. Ng, M. I. Jordan, and Y. Weiss, On spectral clustering: Analysis and an algorithm, in Advances in Neural Information Processing Systems 14: Proceedings of the 2001 Conference (MIT Press, Cambridge, MA, 2002), pp. 849--856]. With roots in graph theory, it uses the spectral properties of the Laplacian matrix to project the data in a low-dimensional space where clustering is more efficient. Despite its success in clustering tasks, spectral clustering suffers in practice from a fast-growing running time of
O(n3), where
n is the number of points in the data set. In this work we propose an end-to-end quantum algorithm performing spectral clustering, extending a number of works in quantum machine learning. The quantum algorithm is composed of two parts: the first is the efficient creation of the quantum state corresponding to the projected Laplacian matrix, and the second consists of applying the existing quantum analog of the
k-means algorithm [I. Kerenidis, J. Landman, A. Luongo, and A. Prakash,
q-means: A quantum algorithm for unsupervised machine learning, in Advances in Neural Information Processing Systems 32: Proceedings of the 2019 Conference (Curran Associates, Red Hook, NY, 2020), pp. 4136--4146]. Both steps depend polynomially on the number of clusters, as well as precision and data parameters arising from quantum procedures, and polylogarithmically on the dimension of the input vectors. Our numerical simulations show an asymptotic linear growth with
n when all terms are taken into account, significantly better than the classical cubic growth. This work opens the path to other graph-based quantum machine learning algorithms, as it provides routines for efficient computation and quantum access to the incidence, adjacency, and projected Laplacian matrices of a graph.
Matrix product states minimize bipartite correlations to compress the classical data representing quantum states. Matrix product state algorithms and similar tools---called tensor network methods---form the backbone of modern numerical methods used to simulate many-body physics. Matrix product states have a further range of applications in machine learning. Finding matrix product states is in general a computationally challenging task, a computational task which we show quantum computers can accelerate. We present a quantum algorithm which returns a classical description of a
k-rank matrix product state approximating an eigenvector given black-box access to a unitary matrix. Each iteration of the optimization requires
O(n⋅k2) quantum gates, yielding sufficient conditions for our quantum variational algorithm to terminate in polynomial-time.
Clustering is one of the most fundamental tasks in machine learning, and the k-means clustering algorithm is perhaps one of the most widely used clustering algorithms. However, it suffers from several limitations, such as sensitivity to centroid initialization, difficulty capturing non-linear structure, and poor performance in high-dimensional spaces. Recent work has proposed improved initialization strategies and quantum-assisted distance computation, but the similarity metric itself has largely remained classical. In this study, we propose a quantum-enhanced variant of k-means that replaces the Euclidean distance with a quantum kernel derived from the inner product between feature-mapped quantum states. Using the Iris dataset, we use multiple quantum feature maps, including entangled SU2 and ZZ circuits, to embed classical data into a higher-dimensional Hilbert space where cluster structures become more separable. We will also be testing using another dataset, namely the breast cancer dataset. Similarity between data points is computed through the inner product between two states. Our results show that this approach achieves improved clustering stability and competitive accuracy compared to the classical algorithm, with the SU2 feature map yielding an accuracy of 88.6 % on the Iris dataset and 91.0 % on the breast cancer dataset, despite operating on NISQ-feasible shallow circuits. These findings suggest that quantum kernels provide a richer similarity landscape than traditional distance metrics, offering a promising path toward more robust unsupervised learning in the NISQ era.