qc-kmeans: A Quantum Compressive K-Means Algorithm for NISQ Devices

Abstract

Clustering on NISQ hardware is constrained by data loading and limited qubits. We present \textbf{qc-kmeans}, a hybrid compressive $k$-means that summarizes a dataset with a constant-size Fourier-feature sketch and selects centroids by solving small per-group QUBOs with shallow QAOA circuits. The QFF sketch estimator is unbiased with mean-squared error $O(\varepsilon^2)$ for $B,S=Θ(\varepsilon^{-2})$, and the peak-qubit requirement $q_{\text{peak}}=\max\{D,\lceil \log_2 B\rceil + 1\}$ does not scale with the number of samples. A refinement step with elitist retention ensures non-increasing surrogate cost. In Qiskit Aer simulations (depth $p{=}1$), the method ran with $\le 9$ qubits on low-dimensional synthetic benchmarks and achieved competitive sum-of-squared errors relative to quantum baselines; runtimes are not directly comparable. On nine real datasets (up to $4.3\times 10^5$ points), the pipeline maintained constant peak-qubit usage in simulation. Under IBM noise models, accuracy was similar to the idealized setting. Overall, qc-kmeans offers a NISQ-oriented formulation with shallow, bounded-width circuits and competitive clustering quality in simulation.

Figures (4)

• Figure 1: Qc-kmeans formulation
• Figure 2: Hadamard-test circuits for estimating the complex quantity $\mu^{(j)}_M=\langle \psi \mid U \mid \psi \rangle$. (a) Real part: the $n_i=3$ index qubits are prepared in a uniform superposition and control the diagonal oracle $U=\mathrm{diag}(e^{i\theta_1},\dots,e^{i\theta_B},1,\dots,1)$. Measuring the ancilla in the $X$ basis yields $\Re\,\mu^{(j)}_M=\mathbb{E}[(-1)^Z]$. (b) Imaginary part: inserting $S^\dagger$ on the ancilla before the final $H$ implements a $Y$-basis measurement, giving $\Im\,\mu^{(j)}_M=\mathbb{E}[(-1)^Z]$ under our sign convention.
• Figure 3: Depth-$p$ per-group QAOA for $D_g=4$. Each layer $\ell$ applies $U_C(\gamma_\ell)$ followed by the XY mixer in two sublayers of adjacent pairs: $(1,2)$ and $(3,4)$, then $(2,3)$. For a complete ring, the pair $(4,1)$ can be implemented with a SWAP network or an additional column.
• Figure 4: Comparison of clustering SSE and computational cost across datasets for two parameterizations: (a) clustering SSE vs. frequency parameter $m$, (b) execution time vs. $m$, (c) clustering SSE vs. peak number of qubits $q_{\mathrm{peak}}$, (d) execution time vs. $q_{\mathrm{peak}}$. For subfigures (c) and (d), only $q_{\mathrm{peak}} \in \{8,9,10,11\}$ are shown.

Theorems & Definitions (27)

• Proposition 4.1: Unbiased QFF estimator
• proof : Proof of Proposition \ref{['prop:unbiased']}
• Lemma 4.1: Shot-noise variance
• proof : Proof of Lemma \ref{['lem:shots']}
• Lemma 4.2: Subsampling variance
• proof : Proof of Lemma \ref{['lem:subsample']}
• Corollary 4.1: MSE decomposition and sample complexity
• proof : Proof of Corollary \ref{['cor:mse']}
• Lemma 4.3: One-hot invariance
• proof : Proof of Lemma \ref{['lem:onehot']}