Differential Privacy of Quantum and Quantum-Inspired Classical Recommendation Algorithms

We study the differential privacy (DP) of the quantum recommendation algorithm of Kerenidis--Prakash and its quantum-inspired classical counterpart. Under standard low-rank and incoherence assumptions on the preference matrix, we show that the randomness already present in the algorithms' measurement/$\ell_2$-sampling steps can act as a privacy-curating mechanism, yielding $(\varepsilon,\delta)$-DP without injecting additional DP noise through the interface. Concretely, for a system with $m$ users and $n$ items and rank parameter $k$, we prove $\varepsilon=\mathcal O(\sqrt{k/n})$ and $\delta= \mathcal O\big(k^2/\min^2\{m,n\}\big)$; in the typical regime $k=\mathrm{polylog}(m,n)$ this simplifies to $\varepsilon=\tilde{\mathcal O}(1/\sqrt n)$ and $\delta=\tilde{\mathcal O}\big(1/\min^2\{m,n\}\big)$. Our analysis introduces a perturbation technique for truncated SVD under a single-entry update, which tracks the induced change in the low-rank reconstruction while avoiding unstable singular-vector comparisons. Finally, we validate the scaling on real-world rating datasets and compare against classical DP recommender baselines.