q-means: A quantum algorithm for unsupervised machine learning (1812.03584v2)

Abstract: Quantum machine learning is one of the most promising applications of a full-scale quantum computer. Over the past few years, many quantum machine learning algorithms have been proposed that can potentially offer considerable speedups over the corresponding classical algorithms. In this paper, we introduce q-means, a new quantum algorithm for clustering which is a canonical problem in unsupervised machine learning. The $q$-means algorithm has convergence and precision guarantees similar to $k$-means, and it outputs with high probability a good approximation of the $k$ cluster centroids like the classical algorithm. Given a dataset of $N$ $d$-dimensional vectors $v_i$ (seen as a matrix $V \in \mathbb{R}^{N \times d})$ stored in QRAM, the running time of q-means is $\widetilde{O}\left( k d \frac{\eta}{\delta^2}\kappa(V)(\mu(V) + k \frac{\eta}{\delta}) + k^2 \frac{\eta^{1.5}}{\delta^2} \kappa(V)\mu(V) \right)$ per iteration, where $\kappa(V)$ is the condition number, $\mu(V)$ is a parameter that appears in quantum linear algebra procedures and $\eta = \max_{i} ||v_{i}||^{2}$. For a natural notion of well-clusterable datasets, the running time becomes $\widetilde{O}\left( k^2 d \frac{\eta^{2.5}}{\delta^3} + k^{2.5} \frac{\eta^2}{\delta^3} \right)$ per iteration, which is linear in the number of features $d$, and polynomial in the rank $k$, the maximum square norm $\eta$ and the error parameter $\delta$. Both running times are only polylogarithmic in the number of datapoints $N$. Our algorithm provides substantial savings compared to the classical $k$-means algorithm that runs in time $O(kdN)$ per iteration, particularly for the case of large datasets.

• The paper introduces q-means, a quantum clustering algorithm designed as a counterpart to the classical k-means algorithm, utilizing Quantum Random Access Memory (QRAM).
• q-means employs quantum techniques for efficient distance estimation, cluster assignment, and centroid updates, aiming for significant computational speedups, particularly a polylogarithmic dependency on dataset size.
• Theoretical analysis shows potential efficiency gains over classical k-means for well-clusterable large datasets, supported by experimental simulations on synthetic and real-world data like MNIST.

Overview of "q-means: A Quantum Algorithm for Unsupervised Machine Learning"

The paper introduces q-means, a quantum algorithm designed to address the clustering problem, a crucial area of unsupervised machine learning. This work proposes a counterpart to the classical k-means algorithm by leveraging the capabilities of quantum computation, specifically by utilizing Quantum Random Access Memory (QRAM). The primary motivation for developing q-means lies in achieving computational efficiency demonstrated through potential quantum speedups, particularly for large datasets where classical algorithms like k-means may encounter scalability issues.

Algorithmic Details

The q-means algorithm mirrors the classical k-means by integrating steps that iteratively seek optimal clustering through assignment and updates of cluster centroids. However, distinctively, q-means employs quantum procedures for:

1. Distance Estimation: Utilizing quantum states to evaluate distances between data points and centroids, achieving efficient estimation without direct computation of Euclidean distance in traditional sense.
2. Cluster Assignment: Quantum minimum finding techniques are applied, allowing efficient determination of the closest centroids in a way that scales better with the dataset size.
3. Centroid Update: Quantum linear algebra techniques facilitate the update of centroids, with norm estimation and tomography providing classical representations needed for subsequent iterations.

Significant attention is devoted to theoretical analysis recording running time, emphasizing polylogarithmic dependency on the number of data points. This demonstrates substantial savings over classical algorithms, where running time typically scales linearly with the dataset size.

Numerical and Theoretical Results

The paper elaborates on aspects such as precision and convergence of q-means, providing guarantees similar to $\delta$-$k$-means—a robustified version of k-means. Furthermore, authors investigate \textit{well-clusterable} datasets (possessing distinct, well-separated clusters) to demonstrate that the quantum algorithm performs favorably under certain structured data scenarios. Experimental simulations corroborate theoretical findings by applying the algorithm to synthetic and real-world datasets (e.g., MNIST), reinforcing its effectiveness and efficiency.

Practical Implications and Future Directions

The proposed q-means algorithm represents a step toward harnessing quantum mechanics for machine learning at scale, promising particularly notable improvements in large-data environments. The potential reduction in computational complexity opens dialog on the integration of quantum algorithms with classical pipelines, suggesting a hybrid quantum-classical paradigm for machine learning tasks.

The discussions in the paper highlight an ongoing need to evaluate q-means on quantum hardware capable of handling the described processes and to explore further optimizations, such as reducing dependency on data conditioning (i.e., condition numbers) and improving robustness in noisy quantum environments.

As quantum technologies mature, algorithms like q-means could redefine computational limits in unsupervised learning, necessitating ongoing research in quantum algorithmics, error mitigation techniques, and hardware-efficient implementations.