Mathematical comparison of classical and quantum mechanisms in optimization under local differential privacy

Abstract

Let $\epsilon >0$. An $n$-tuple $(p_i{)}_{i=1}^n$ of probability vectors is called $\epsilon$-differentially private ($\epsilon$-DP) if $e^{\epsilon }{p}_j-p_i$ has no negative entries for all $i,j=1,\dots ,n$. An $n$-tuple $({\rho }_i{)}_{i=1}^n$ of density matrices is called classical-quantum $\epsilon$-differentially private (CQ $\epsilon$-DP) if $e^{\epsilon }{\rho }_j-{\rho }_i$ is positive semi-definite for all $i,j=1,\dots ,n$. Denote by ${\mathrm{C}}_n(\epsilon )$ the set of all $\epsilon$-DP $n$-tuples, and by ${\mathrm{CQ}}_n(\epsilon )$ the set of all CQ $\epsilon$-DP $n$-tuples. By considering optimization problems under local differential privacy, we define the subset ${\mathrm{EC}}_n(\epsilon )$ of ${\mathrm{CQ}}_n(\epsilon )$ that is essentially classical. Roughly speaking, an element in ${\mathrm{EC}}_n(\epsilon )$ is the image of $(p_i{)}_{i=1}^n\in {\mathrm{C}}_n(\epsilon )$ by a completely positive and trace-preserving linear map (CPTP map). In a preceding study, it is known that ${\mathrm{EC}}_2(\epsilon )={\mathrm{CQ}}_2(\epsilon )$. In this paper, we show that ${\mathrm{EC}}_n(\epsilon )\ne {\mathrm{CQ}}_n(\epsilon )$ for every $n\ge 3$, and estimate the difference between ${\mathrm{EC}}_n(\epsilon )$ and ${\mathrm{CQ}}_n(\epsilon )$ in a certain manner.

26 pages

References in corpus (4)

• Quantum noise protects quantum classifiers against adversaries
• Quantum Differentially Private Sparse Regression Learning
• A unifying framework for differentially private quantum algorithms
• Bridging Quantum Computing and Differential Privacy: Insights into Quantum Computing Privacy

Cited by in corpus (1)

• Quantum Local Differential Privacy and Quantum Statistical Query Model