Differential Privacy of Quantum and Quantum-Inspired-Classical Recommendation Algorithms

TL;DR

This work studies the differential privacy properties of the quantum recommendation algorithm and its quantum-inspired classical counterpart. It develops a novel low-rank perturbation framework for SVD and a semi-random eigenvector model to enable rigorous DP analysis, proving that for typical users the algorithms satisfy $(\tilde{\mathcal{O}}(1/n), \; \tilde{\mathcal{O}}(1/\min\{m,n\}))$-DP (with explicit bounds for γ-typical users). It also reveals a passive DP phenomenon: the quantum (and quantum-inspired) computations inherently curate privacy without external noise, with the quantum variant offering stronger privacy potential than the classical one. Together, these results provide privacy guarantees for quantum-assisted recommenders and offer guidance for designing privacy-conscious large-scale recommendation systems.

Abstract

We analyze the DP (differential privacy) properties of the quantum recommendation algorithm and the quantum-inspired-classical recommendation algorithm. We discover that the quantum recommendation algorithm is a privacy curating mechanism on its own, requiring no external noise, which is different from traditional differential privacy mechanisms. In our analysis, a novel perturbation method tailored for SVD (singular value decomposition) and low-rank matrix approximation problems is introduced. Using the perturbation method and random matrix theory, we are able to derive that both the quantum and quantum-inspired-classical algorithms are $\big(\tilde{\mathcal{O}}\big(\frac 1n\big),\,\, \tilde{\mathcal{O}}\big(\frac{1}{\min\{m,n\}}\big)\big)$-DP under some reasonable restrictions, where $m$ and $n$ are numbers of users and products in the input preference database respectively. Nevertheless, a comparison shows that the quantum algorithm has better privacy preserving potential than the classical one.

Figures (6)

• Figure 1: An illustration of the low-rank recommendation algorithm process implied by the Eckart-Young theorem.
• Figure 2: Distribution of components of $U_i$ and $V_j$ on some real-world data. Left: Histogram of $u_{\ell i}=(U_i)_\ell$ of the MovieLens dataset_movielens dataset. Due to hardware limitations, dataset is subsampled by a factor of 10 and only the 500 components are computed. Right: Histogram of $v_{\ell j}=(V_j)_\ell$ of a large matrix converted from a $1001\times 1920$ image(see appendix \ref{['appendix:dataset_info']} for more details). Curves are corresponding p.d.f. of distribution $S\mathrm{Proj}(m),S\mathrm{Proj}(n)$ respectively. Note that $p_{S\mathrm{Proj}(n)}(\cdot)$ is quite similar to a normal distribution for large $n$. While $u_{\ell i}$ and $v_{\ell j}$ aren't obeying $S\mathrm{Proj}$ perfectly, the proposed distribution seems to be a good approximation.
• Figure 3: A diagram illustrating our proof strategy.
• Figure 4: Demonstration of the core lemma. $\delta(k)$, $f(k)$ and $95\%$ probability bound by core lemma on some real-world data is computed and plotted. Left: MovieLens dataset_movielens dataset. Due to hardware limitations, MovieLens dataset is subsampled by a factor of 10 and only first 500 singular terms are computed. Right: An $1001\times 1920$ image(the same image as in figure \ref{['fig:uv_stat']}). For small $k$, $\delta(k)$ appears to be linear and close to $f(k)$; For large $k\sim m,n$, $\delta(k)$ saturates to $1$ and grows slower than $f(k)$. Because full-scale SVD is required to do the computation and recommendation databases are typically extremely large and sparse(see table \ref{['table:dataset_stats']}), we are unable to extend our experiment to larger datasets.
• Figure 5: Global norm change of classical algorithm output ($|\Delta_{\le k}|$, purple) and quantum algorithm output ($|(\Delta_{ \le k})_i\,|=|\left|(T'_{\le k})_i\right\rangle-\left|(T_{\le k})_i\right\rangle|$ on different data, orange) when changing one element in database, as a function of $k$. $\delta(k)$ is also plotted as a reference. The classical curve(purple) exhibits unstable behaviour, has higher value and is hard to bound compared to the quantum curve(orange). Left: MovieLens dataset dataset_movielens; Right: An $1001\times 1920$ image (the same image as in figure \ref{['fig:main_thm']}).

Theorems & Definitions (15)

• Theorem 1: Main Result Preview, Informal
• Definition 1: $\ell^2$-norm sampling
• Definition 2: Amplitude Encoding
• Theorem 2: Eckart-Youngeckart-young, Best low-rank approximation
• Definition 3: Simplified Recommendation Algorithms
• Definition 4: Differential Privacydwork_algodpbook
• Theorem 3: Marcenko-Pasturfeier_randmat, Singular Value Distribution of Noise
• Theorem 4: Singular Vector Distribution of Wishart Matrices wishart_rand_mat
• Definition 5: Semi Random Eigenvector Condition
• Theorem 5: Properties of $\mathrm{Uniform}(S^{N-1})$