Quantum K-Means Algorithmic Frameworks

• Quantum k-means algorithmic frameworks are a diverse set of methods that recast classical k-means using quantum subroutines and hybrid quantum–classical updates.
• They employ key techniques such as quantum distance estimation, minimum finding, and specialized embedding to overcome classical computational bottlenecks.
• These methods promise exponential speedups under QRAM models while offering NISQ-compatible approximations and rigorous guarantees on clustering performance.

Quantum k-means algorithmic frameworks constitute a diverse set of methodologies for accelerating or enhancing Lloyd-style clustering with quantum, hybrid quantum–classical, or quantum-inspired algorithms. These frameworks address classical computational bottlenecks through a spectrum of circuit-based, annealing-based, variational, and compressed-feature approaches, and range from theoretically exponential quantum speedups (conditional on QRAM) to NISQ-compatible, quantum-inspired approximations. Central technical elements include quantum distance estimation subroutines, fast minimum search routines, specialized embedding and kernel constructions, and rigorous analysis of both complexity and robustness.

1. Theoretical Mapping of k-Means to the Quantum Regime

The classical k-means objective seeks $$\arg\min_{\{C_j\},\{\omega_j\}} \sum_{j} \sum_{x_i \in C_j} \|x_i - \omega_j\|^2$$ through alternating assignment (nearest centroid) and update (mean of cluster) steps. Quantum k-means frameworks recast this process:

• Quantum distance computation replaces classical Euclidean distances with quantum overlap, typically via the SwapTest or variants; for n-dimensional unit vectors $\ket{x},\ket{\omega}$, SwapTest yields $$P(0) = \frac{1}{2} + \frac{1}{2}|\langle x|\omega\rangle|^2$$, enabling estimation of $\|x-\omega\|^2$ up to quadratic corrections (Gong et al., 2020, Khan et al., 2019, Modi et al., 2023).
• Quantum minimum finding leverages algorithms such as GroverOptim or the Dür–Høyer scheme: given a black-box table of distances, the minimum can be found in $O(\sqrt{k})$ (for $k$ centroids) quantum queries (Gong et al., 2020, Kerenidis et al., 2018).
• The overall structure alternates quantum assignment/labeling with (typically) classical centroid updates or quantum linear-algebraic procedures for state-based centroid recomputation (Kerenidis et al., 2018, Doriguello et al., 2023).

In matrix product state (MPS)–based quantum-inspired methods, classical data are first lifted into a tensorized Hilbert space (MPS with bond dimension $\chi$), and the centroid is optimized variationally in this enlarged space to improve cluster separation and escape local minima (Shi et al., 2020).

2. Major Quantum k-Means Frameworks

A broad inventory of frameworks includes:

Framework	Main Quantum Subroutines	Scaling Behavior (Per Iteration)
Quantum Lloyd/q-means (Kerenidis et al., 2018, Doriguello et al., 2023)	Quantum distance estimation, minimum finding, amplitude estimation, QRAM	$O(\mathrm{poly}(k,d,\cdots)\log N)$ (polylog in N)
Quantum k-means with QHE (Gong et al., 2020)	SwapTest, GroverOptim, quantum homomorphic encryption, T-gate update with trusted servers	$O(M \log n \sqrt{k} \cdot t)$
MPS quantum-inspired (Shi et al., 2020)	MPS embedding, Hilbert-space loss, variational MPS sweep	$O(NK d \chi^2 + K d (\chi^3+N\chi^2))$
Hybrid cluster assignment (Poggiali et al., 2022, Khan et al., 2019, Modi et al., 2023)	Parallelized quantum distance, amplitude/kernels, destructive/constructive interference	$O(M)$ to $O(kM)$ assignments
Quantum $D^2$-sampling / $k$-means++ (Shah et al., 2024)	QRAM, swap test, amplitude estimation, rejection sampling	$\tilde{O}(\zeta^2 k^2)$, $O(\log N)$ dependence
Variational eigensolver + Coreset (Yung et al., 2023)	Ising Hamiltonian mapping, VQE with contour coresets	Qubits $\sim$ coreset size, circuit depth $\sim$ ansatz layers

Each framework addresses specific algorithmic phases (initialization, assignment, update) and system constraints (device width/depth, data access models, robustness against imperfections).

3. Key Algorithmic Components

Quantum Distance Estimation

• SwapTest: Core quantum subroutine for overlap estimation, requiring an ancilla, amplitude- or angle-encoding for vector preparation, and control-SWAP operations (Khan et al., 2019, Gong et al., 2020, Poggiali et al., 2022). Alternative circuits leverage destructive/constructive interference with or without ancilla, with shallowest depth realized in negative-rotation schemes for angle-encoded data (Khan et al., 2019).
• Kernel-based approaches: Map classical data into quantum states and use kernel metrics such as the squared fidelity $|\langle \psi(x)|\psi(c)\rangle|^2$ as a clustering surrogate, forming a quantum or "quantum-inspired" kernel k-means (Oswal et al., 23 Sep 2025, Modi et al., 2023).

Quantum Minimum Finding

• GroverOptim/Quantum Minimum-Finding: Uniform superposition over candidate centroids, phase-flip oracles based on comparison to current threshold, inversion-about-the-mean diffusion, iterative measurement, and classical update (Gong et al., 2020). Amplitude amplification is used for quadratic speedup in minimal distance search.
• Hybrid parallel assignment: By increasing circuit width, multiple record–centroid distances can be computed in quantum parallel, achieving assignment in constant time for q$_{M:k}$ schemes at the expense of QRAM overhead and shot complexity (Poggiali et al., 2022).

Randomized/Compressed Sampling

• Uniform mini-batch quantum k-means: Replaces full-data assignment/update with uniform or $D^2$ quantum sampling, leveraging amplitude amplification and quantum mean estimation to approximate Lloyd-steps with sample-size bounds depending on intrinsic within-cluster variance $\varphi$ (Chen et al., 29 Apr 2025, Shah et al., 2024).
• Quantum compressive k-means (qc-kmeans): Compresses large datasets to a fixed-size random Fourier feature quantum sketch, performs per-group QUBO optimization of centroids via depth-1 QAOA, and iterates with elitist retention (Chumpitaz-Flores et al., 26 Oct 2025).

Adiabatic and Variational Techniques

• QUBO/Ising mapping and quantum annealing: The balanced $k$-means cost is formulated as a QUBO, mapped via binary variables to Ising Hamiltonians, then minimized via adiabatic evolution on D-Wave hardware. Penalties enforce assignment and cluster size constraints (Arthur et al., 2020).
• VQE with quantum-tailored coreset: Reduces large datasets to a contour coreset, constructs a weighted Ising Hamiltonian for coreset clustering, then optimizes assignment via VQE on shallow, few-qubit circuits (Yung et al., 2023).

4. Complexity, Rigorous Guarantees, and Practical Implementations

Complexity and Speedup

• Exponential-in-N savings: QRAM-based frameworks (q-means, $D^2$-sampling) achieve per-iteration cost only polylogarithmic in dataset size $N$, with polynomial dependence on $k,d$ and data-dependent parameters ($\eta$, aspect ratio, condition number) (Kerenidis et al., 2018, Doriguello et al., 2023, Shah et al., 2024, Jaiswal, 2023).
• Approximation guarantees: $(1+\varepsilon)$-approximation quantum schemes (Jaiswal, 2023), quantum $k$-means++ (Shah et al., 2024) and quantum uniform mini-batch (Chen et al., 29 Apr 2025) all provide certified bounds (relative to global optimum).
• Quantum-inspired classical parity: Multiple frameworks, notably dequantized $q$-means and QI-$k$-means++, demonstrate classical algorithms with the same $O(\log N)$ scaling, but with worse polynomial factors (Doriguello et al., 2023, Shah et al., 2024).

Empirical Results and Hardware Constraints

• NISQ-tailored circuits: Negative-rotation and destructive-interference schemes achieve perfect clustering at circuit depths of 2–14 gates and 3–5 qubits for low-dimensional data (Khan et al., 2019).
• Hybrid quantum-classical loops: Assignments via quantum or kernel-based subroutines, update via classical centroid recomputation (Poggiali et al., 2022, Modi et al., 2023).
• Quantum cloud and security: Homomorphic encryption (QHE) and trusted-server key management enable delegated privacy-preserving quantum k-means on cloud hardware, confirmed on IBM Qiskit simulators (Gong et al., 2020).

5. Quantum Embedding and Data Preparation

• Vector encoding: Feature vectors are amplitude-encoded ($|x\rangle$) or angle-encoded ($\prod_j R_y(\theta_j)|0\rangle$) for SwapTest or kernel methods; bond-dimension $\chi$ controls entanglement in MPS-based approaches (Shi et al., 2020, Oswal et al., 23 Sep 2025).
• Quantum kernels: Overlaps between encoded states serve as quantum kernels, with the embedding choice directly affecting the metric properties and clustering performance, especially under data with amplitude-phase noise (Modi et al., 2023, Oswal et al., 23 Sep 2025).
• Data access limitations: The QRAM model underlies most exponential speedup claims but remains a significant bottleneck for physical implementation. Data loading costs $O(Nd)$ dominate unless quantum-state data is naturally available (Kerenidis et al., 2018, Shah et al., 2024).

6. Robustness, Limitations, and NISQ/Cloud Considerations

• Noise and error mitigation: SwapTest and similar circuits are robust at low depth but degrade with circuit depth and shot noise; negative-rotation schemes are robust for angular data under decoherence (Khan et al., 2019). Quantum cloud protocols support encrypted computation but introduce T-gate key update cost and require trust split between servers (Gong et al., 2020).
• Scalability bottlenecks: FF-QRAM and post-selection exponentially suppress parallel speedup as data or circuit width increases (Poggiali et al., 2022). QUBO embedding for $Nk$ logical variables is a limiting factor for quantum annealing scaling (Arthur et al., 2020). NISQ devices limit circuit depth and favor per-group QUBOs and compressive surrogates (Chumpitaz-Flores et al., 26 Oct 2025).
• Quantum-inspired improvements: MPS-based and minimalistic kernel approaches show that classical simulations of quantum cleverness (encoding, initialization, kernel design) can outperform baseline k-means for various ARI/Silhouette metrics, e.g., on Iris and Seeds (Shi et al., 2020, Oswal et al., 23 Sep 2025).

7. Outlook and Open Problems

• Full quantum end-to-end pipelines remain a challenge due to bottlenecks in data loading, centroid update, and high-dimensional state preparation.
• Hybrid and modular strategies—splitting quantum assignment and classical update, exploiting quantum-inspired kernels, and leveraging coresets—currently provide the most viable path toward near-term hardware execution.
• Rigorous analysis of approximation ratios and robustness to data-dependent degeneracies (aspect ratio, within-cluster variance, cluster imbalance) is active, with uniform quantum sampling providing parameter-improved guarantees (Chen et al., 29 Apr 2025, Jaiswal, 2023).
• Kernel-method parity: The equivalence of quantum distance surrogates to reproducing kernel Hilbert space clustering implies that many quantum subroutines function as kernel tricks; establishing clear separation from classical kernel-k-means is necessary for isolating genuine quantum advantage (Modi et al., 2023, Oswal et al., 23 Sep 2025).

In sum, quantum k-means algorithmic frameworks integrate quantum distance kernels, fast minimum search, advanced data encoding, and, in several schemes, privacy or compressive surrogates. Their theoretical complexity, approximation guarantees, implementation trade-offs, and empirical benchmarking establish a taxonomy spanning provable exponential quantum speedups (conditional on QRAM and circuit depths) to NISQ-efficient quantum-inspired clustering methods (Gong et al., 2020, Kerenidis et al., 2018, Doriguello et al., 2023, Chen et al., 29 Apr 2025, Shi et al., 2020, Poggiali et al., 2022, Chumpitaz-Flores et al., 26 Oct 2025, Khan et al., 2019, Modi et al., 2023, Yung et al., 2023, Shah et al., 2024, Jaiswal, 2023, Oswal et al., 23 Sep 2025, Arthur et al., 2020).