Quantum K-Means Clustering

• Quantum K-Means is a quantum-enhanced clustering method that integrates quantum state encoding and parallelism to improve distance estimation and centroid updates.
• It replaces core k-means steps with quantum subroutines like the swap test and quantum kernel methods, achieving asymptotic speedups and enhanced clustering accuracy.
• Practical implementations address challenges such as state preparation, error mitigation, and resource trade-offs while advancing quantum-inspired algorithms for real-world data.

A Quantum K-Means algorithm is a family of algorithmic frameworks that apply quantum computing and quantum-inspired techniques to accelerate or enhance the standard k-means clustering method, which partitions unlabeled data into clusters such that each point is assigned to the nearest centroid. Quantum k-means approaches leverage principles such as quantum state encoding, quantum distance estimation, amplitude amplification, parallelism, or variational optimization in a high-dimensional Hilbert space to either accelerate core subroutines or to introduce fundamentally new clustering mechanisms. These methods have been evaluated in classical simulation, on quantum hardware, and in quantum-inspired classical frameworks, with performance measured in terms of computational runtime, clustering accuracy, scalability, and stability of clustering assignments.

1. Quantum K-Means Algorithmic Frameworks

Quantum k-means algorithms generalize the classical k-means procedure by replacing one or more key steps—distance estimation, cluster assignment, or centroid update—with quantum or quantum-inspired subroutines. Foundational approaches include:

• Quantum Distance Estimation: Quantum circuits (e.g., swap test) are used to compute the closeness between high-dimensional data points and centroids, replacing the O(d) or O(dN) cost of classical vector arithmetic with quantum sampling or interference that can achieve effective O(log d) resource scaling (Kerenidis et al., 2018, Sarma et al., 2019).
• Quantum Kernel and State Overlap: Rather than assigning clusters by minimizing Euclidean distance, some approaches measure the overlap between quantum-encoded data states and centroid states, using quantum kernels K(x, c) = |⟨ψ(c)|ψ(x)⟩|². This indirectly computes a quantum distance metric (Oswal et al., 23 Sep 2025).
• Amplitude Encoding and Parallelism: High-dimensional vectors are encoded into superpositions over O(log N) qubits, enabling quantum circuits to compute multiple distances or assignments in parallel. Various levels of quantum parallelism exist, ranging from single pairwise comparisons to global parallel assignment of all data points (Poggiali et al., 2022).
• Quantum-Inspired Classical Algorithms: Techniques inspired by quantum encoding (e.g., matrix product state representations or stereographic projection onto the Bloch sphere) yield more expressive or robust clustering in classical domains (Shi et al., 2020, Jasso et al., 2023).

2. Key Quantum Subroutines and Encoding Strategies

The efficiency and accuracy of quantum k-means rely critically on the choice and implementation of state preparation and quantum arithmetic, as evidenced by multiple approaches:

• State Preparation: Data vectors are encoded as quantum states by either amplitude encoding (logarithmic number of qubits), angle encoding (component-wise rotations), or hybrid schemes. Amplitude encoding increases expressivity but is more demanding in terms of state preparation and gate depth (Oswal et al., 23 Sep 2025, Poggiali et al., 2022).
• Swap Test and Interference: The quantum swap test is widely used to estimate the inner product between two quantum states, yielding a probability whose deviation from 1/2 is proportional to their squared Euclidean distance (or, for angle encoding, to a cosine kernel) (Sarma et al., 2019, DiAdamo et al., 2021).
• Quantum Kernel Methods: Instead of explicit distances, the inner product or kernel overlap between encoded data and centroid states determines cluster assignment. Quantum kernel methods may offer distinct partitionings compared to classical Euclidean metrics, particularly with hybrid or minimally entangled encoding (Oswal et al., 23 Sep 2025).
• Stereographic Projection for Quantum Embedding: For two-dimensional data, stereographic projection maps planar points onto the Bloch sphere, enabling quantum measurement of cosine similarity or fidelity directly via Bell-state circuits, increasing accuracy and convergence speed compared to amplitude or naive angle encoding (Jasso et al., 2023).

Table: Example Encoding Strategies Employed in Quantum K-Means

Encoding Scheme	Qubits Required	Quantum Feature
Angle Encoding	O(d)	Shallow, minimal entanglement
Amplitude Encoding	log₂(d)	High entanglement, compact
Hybrid Encoding	mix of O(d) and log₂(d)	Feature-weighted flexibility
Stereographic/Bloch	1 (for 2D data)	Fidelity/cosine dissimilarity

3. Runtime Efficiency, Scalability, and Complexity

Quantum k-means algorithms can achieve significant asymptotic improvements over classical k-means for large-scale clustering, though performance depends on the algorithmic instantiation and device characteristics:

• Asymptotic Speedups: Algorithms leveraging quantum RAM (QRAM) to access data achieve polylogarithmic scaling in the number of data points N, reducing core per-iteration costs from O(kdN) classically to Õ(kd(η/δ²)κ(V)[μ(V)+k(η/δ)]) (where κ(V) and μ(V) are data-dependent, η is squared norm bound, and δ precision) (Kerenidis et al., 2018). Simpler quantum-inspired methods can reach O(Mk log N) scaling for M input vectors and k clusters (Khan et al., 2019). For well-clusterable datasets, complexity can approach linear in d and polynomial in k, with only polylogarithmic dependence on N.
• Quantum Approximation Schemes: Recent work achieves (1+ε)-approximation in time Õ(2^{O(k/ε)}dζ²) where ζ is the aspect ratio, again with only polylog(N) dependence (Jaiswal, 2023, Shah et al., 22 May 2024).
• Hardware and Practical Factors: Direct speedup depends not just on circuit complexity but on state preparation overhead, coherence times, and error rates; simulation-based results often show significant classical overheads unless circuits are kept minimal (shallow, with post-selection and parallelization where practical) (DiAdamo et al., 2021, Poggiali et al., 2022).

4. Clustering Accuracy, Robustness, and Enhanced Metrics

Quantum k-means and quantum-inspired methods can yield accuracy and performance competitive with, or superior to, classical k-means in various regimes:

• Accuracy Benchmarks: On datasets such as Iris, Wine, and real-world telecommunication data, quantum k-means implementations achieve results comparable to classical methods, with ARI sometimes improved by quantum or quantum-inspired encoding (e.g., a hybrid encoding achieves ARI ≈ 0.88 on Iris v. 0.70 for classical) (Oswal et al., 23 Sep 2025).
• Robustness to Initialization and Local Minima: Quantum-influenced centroid selection and variational optimization via MPS (matrix product states) can increase both convergence speed and robustness to local minima (Shi et al., 2020).
• Effect of Quantum Distances and Kernels: Quantum overlap and fidelity-based metrics introduce alternative cluster geometries, potentially capturing data relationships missed by Euclidean metrics—particularly in high-noise, nonlinear, or overlapping cluster regimes (Jasso et al., 2023).
• Soft Assignments and Interference Effects: In quantum-inspired GMM variants, destructive interference leads to sharper class boundaries, yielding lower parameter estimation errors, order-of-magnitude improvements in center estimation, and enhanced resistance to class overlap and non-ideal Gaussianity (Rahman et al., 2016).

5. Implementation Considerations and Quantum Hardware

The practical realization of quantum k-means depends on hardware resource constraints, error mitigation, and efficient classical-quantum hybrid strategies:

• Circuit Depth and Post-Selection: Short-depth quantum circuits are crucial on NISQ devices; approaches based on negative rotations or minimalistic encoding yield near-classical accuracy for small qubit numbers with as few as 2–14 gates (Khan et al., 2019, Poggiali et al., 2022).
• Noise and Error Mitigation: Quantum AutoEncoders and noise-aware coreset selection are explored to maintain clustering accuracy in the presence of depolarization and bit-flip errors, though such techniques can introduce additional resource costs (Qu et al., 2022).
• Coreset Selection and Data Compression: Coreset construction (BFL16, ONESHOT) allows quantum circuits to operate on weighted subsets, trading off between clustering fidelity and manageable circuit sizes, especially for QAOA-based clustering (Qu et al., 2022).
• Hybrid Cloud Quantum Algorithms: Cloud-delegated implementations with encrypted quantum data (using quantum homomorphic encryption and trusted execution for T-gate updates) permit secure, outsourced quantum k-means processing, with the actual quantum subroutines (swap test, GroverOptim) running remotely and only decryption and centroid re-computation on the client side (Gong et al., 2020).

6. Relation to Quantum Spectral Clustering and Other Paradigms

Quantum k-means is only one member of a broader class of quantum clustering algorithms. Quantum spectral clustering, for example, leverages quantum phase estimation to expose graph Laplacian structure, supplementing quantum k-means where the latter fails for nonconvex or non-Euclidean data geometries (Kerenidis et al., 2020, Li et al., 2022). In contrast to direct assignment via quantum minimum finding or swap testing, spectral methods exploit eigenvectors in projected Hilbert spaces and complement the geometric reach of distance-based clustering.

7. Limitations, Open Problems, and Future Directions

• Scalability and Resource Trade-Offs: While quantum speedup is asymptotically favorable as N grows, bottlenecks remain in QRAM construction, state preparation, and hybrid circuit-classical iteration overhead.
• Encoding Selection and Quantum Metrics: Determining empirically and theoretically which encoding (angle, amplitude, hybrid, stereographic) best suits various data structures, as well as the optimal choice of quantum distance or kernel, remains an active line of research.
• Noise, Error Mitigation, and Real Devices: Error rates and decoherence in NISQ devices limit the depth and fidelity of quantum clustering circuits. Continued development of shallow, robust circuits, together with error-mitigation strategies, is necessary for practical deployment.
• Quantum-Inspired Algorithms and Dequantization: Many procedures originally thought to be uniquely quantum have inspired classical analogues with similar asymptotics in the sample-query access model; understanding the unique contributions and boundaries of true quantum speedup is a current research question.
• Integration with Advanced Clustering Paradigms: Current focus is sharpening the quantum (or quantum-inspired) enhancement in complex clustering setups, such as balanced clustering, fuzzy assignment extensions, and graph-based community detection.

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Quantum k-means represents a rich and evolving intersection of quantum computing, linear algebra, and unsupervised machine learning; the field now spans from foundational algorithmic speedups and encoding techniques to robust error-aware implementations and quantum-inspired classical frameworks. Its ongoing development is informed by advances in quantum hardware, theoretical algorithmics, and the practical needs of large-scale scientific data analysis.