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arXiv:2011.09960 · Journal of Physics A Mathematical and Theoretical

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

Yuuya Yoshida

Published Dec 18, 2024• 4 citations
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Abstract

Abstract Differential privacy (DP) is an influential method of protecting private data, and is a condition for a conditional probability distribution. Since we can regard DP as a condition for a tuple of probability vectors, it is natural to consider a similar condition for a tuple of density matrices as a quantum version of DP. This condition is called classical-quantum DP (CQ-DP) because it is considered in converting classical data to quantum states. In the study of DP (including CQ-DP), a positive parameter ɛ represents the privacy level to be guaranteed. In this paper, we show that CQ-DP has a quantum advantage in certain optimization problems. Moreover, we compare classical DP and CQ-DP mathematically to clarify a relation between privacy level and quantum advantage.

DOI
10.1088/1751-8121/ada0f9

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arXiv:2403.09173•IEEE International Conference on Quantum Computing and Engineering (QCE)•Sep 15, 2024•5 citations
Abstract Differential privacy (DP) is an influential method of protecting private data, and is a condition for a conditional probability distribution. Since we can regard DP as a condition for a tuple of probability vectors, it is natural to consider a similar condition for a tuple of density matrices as a quantum version of DP. This condition is called classical-quantum DP (CQ-DP) because it is considered in converting classical data to quantum states. In the study of DP (including CQ-DP), a positive parameter ɛ represents the privacy level to be guaranteed. In this paper, we show that CQ-DP has a quantum advantage in certain optimization problems. Moreover, we compare classical DP and CQ-DP mathematically to clarify a relation between privacy level and quantum advantage.
Differential privacy (DP) is an influential privacy measure and has been studied to protect private data. DP has been often studied in classical probability theory, but few researchers studied quantum versions of DP. In this paper, we consider classical-quantum DP mechanisms which (i) convert binary private data to quantum states and (ii) satisfy a quantum version of the DP constraint. The class of classical-quantum DP mechanisms contains classical DP mechanisms. As a main result, we show that some classical DP mechanism optimizes any information quantity satisfying the information processing inequality. Therefore, the performance of classical DP mechanisms attains that of classical-quantum DP mechanisms.
We optimize the trade-off between privacy and utility in the high-privacy regime. We adopt local differential privacy (LDP) and its quantum extension, quantum local differential privacy (QLDP), for privacy protection, and investigate utility functions including the Holevo information (which reduces to the mutual information in the classical case) and the error exponents in symmetric and asymmetric hypothesis testing. Under the LDP and QLDP constraints, these utility functions have classical and quantum optimal values, which we denote simply by CCC and QQQ, respectively, in this abstract. In this paper, we provide optimal LDP and QLDP mechanisms achieving the classical and quantum optimal values in the high-privacy regime, and prove that the asymptotic ratio Q/CQ/CQ/C in this regime takes the same value regardless of the utility function. Our results reveal quantum advantages (specifically, Q/C≥3/2Q/C\ge3/2Q/C≥3/2) for the above utility functions when the private data take at least three possible values. In addition, conditional on a mathematical conjecture concerning equi-isoclinic tight fusion frames (EITFFs), our QLDP mechanisms also exhibit quantum advantages in a moderate-privacy regime.
Let ε>0\varepsilon>0ε>0. An nnn-tuple (pi)i=1n(p_i)_{i=1}^n(pi​)i=1n​ of probability vectors is called (classical) ε\varepsilonε-differentially private (ε\varepsilonε-DP) if eεpj−pie^\varepsilon p_j-p_ieεpj​−pi​ has no negative entries for all i,j=1,…,ni,j=1,\ldots,ni,j=1,…,n. An nnn-tuple (ρi)i=1n(\rho_i)_{i=1}^n(ρi​)i=1n​ of density matrices is called classical-quantum ε\varepsilonε-differentially private (CQ ε\varepsilonε-DP) if eερj−ρie^\varepsilon\rho_j-\rho_ieερj​−ρi​ is positive semi-definite for all i,j=1,…,ni,j=1,\ldots,ni,j=1,…,n. We denote by Cn(ε)\mathrm{C}_n(\varepsilon)Cn​(ε) the set of all ε\varepsilonε-DP nnn-tuples, by CQn(ε)\mathrm{CQ}_n(\varepsilon)CQn​(ε) the set of all CQ ε\varepsilonε-DP nnn-tuples, and by ECn(ε)\mathrm{EC}_n(\varepsilon)ECn​(ε) the set of all nnn-tuples (∑kpi(k)σk)i=1n(\sum_k p_i(k)\sigma_k)_{i=1}^n(∑k​pi​(k)σk​)i=1n​ with (pi)i=1n∈Cn(ε)(p_i)_{i=1}^n\in\mathrm{C}_n(\varepsilon)(pi​)i=1n​∈Cn​(ε) and density matrices σk\sigma_kσk​. The set ECn(ε)\mathrm{EC}_n(\varepsilon)ECn​(ε) is a subset of CQn(ε)\mathrm{CQ}_n(\varepsilon)CQn​(ε), and is essentially classical in optimization. In a preceding study, it is known that EC2(ε)=CQ2(ε)\mathrm{EC}_2(\varepsilon)=\mathrm{CQ}_2(\varepsilon)EC2​(ε)=CQ2​(ε). In this paper, we show that ECn(ε)≠CQn(ε)\mathrm{EC}_n(\varepsilon)\not=\mathrm{CQ}_n(\varepsilon)ECn​(ε)=CQn​(ε) for every n≥3n\ge3n≥3, and give a sufficient condition for a CQ ε\varepsilonε-DP nnn-tuple not to lie in ECn(ε)\mathrm{EC}_n(\varepsilon)ECn​(ε).
Differential privacy (DP) is a widely used framework for protecting sensitive information in data analysis and machine learning. With growing interest in quantum computing, substantial effort has gone into extending DP to quantum algorithms. However, many existing formulations emphasise global state distinguishability or privacy for channels that output quantum states, which is often misaligned with settings where the final output consists of classical measurement data, as in both fault-tolerant and near-term devices. In this work, we develop an observable-based approach to quantum DP tailored to expectation-value estimation from quantum measurements. We introduce a neighbouring relation defined directly in terms of observable expectation values and give general private mechanisms for broad measurement protocols, including classical shadows and eigenbasis measurements. We also show that realistic device noise (e.g., local depolarising and generalised amplitude damping) can amplify privacy guarantees for k -local observables. Complementing these positive results for private measurements, we establish limitations for differentially private mechanisms that release quantum states. Finally, we show that our neighbouring relation is compatible with quantum encodings of classical data, enabling privacy guarantees for the underlying classical inputs.
With the emergence of machine learning (ML) models trained on large datasets containing potentially sensitive data, a major question in AI safety is how to make learning private with respect to the training data. Similar to classical machine learning, quantum machine learning (QML) models are not devoid of privacy vulnerabilities. Differential privacy (DP) is a standard tool for training ML models on sensitive data, but its impact in QML remains poorly understood. In this work we study private training in hybrid variational QML models using a classical private DP-SGD optimizer applied to pipelines with classical inputs and outputs. We analyze the interplay between gradient clipping and calibrated noise addition in DP-SGD, and its impact on optimization and accuracy for noisy and noiseless quantum models. We first explain why quantum noise does not provide a satisfactory replacement for the calibrated noise in DP-SGD for ensuring privacy. We then show how the deterministic bounds on gradient norms for a wide class of quantum models translate into explicit control of the detrimental clipping bias introduced by DP-SGD. Finally, we formulate a numerical comparison protocol under fixed clipping threshold and privacy budget and evaluate it on synthetic and image-classification tasks for equivalent quantum and classical models. Our results suggest that quantum models can retain higher accuracy in private-training regimes where the formal privacy guarantee is ensured by a classical DP-SGD mechanism.
While quantum computing has strong potential in data-driven fields, the privacy issue of sensitive or valuable information involved in the quantum algorithm should be considered. Differential privacy (DP), which is a fundamental privacy tool widely used in the classical scenario, has been extended to the quantum domain, i.e., quantum differential privacy (QDP). QDP may become one of the most promising approaches toward privacy-preserving quantum computing since it is not only compatible with classical DP mechanisms but also achieves privacy protection by exploiting unavoidable quantum noise in noisy intermediate-scale quantum (NISQ) devices. This paper provides an overview of the various implementations of QDP and their performance in terms of privacy parameters under the DP setting. Specifically, we propose a taxonomy of QDP techniques, categorizing the literature on whether internal or external randomization is used as a source to achieve QDP and how these implementations are applied to each phase of the quantum algorithm. We also discuss challenges and future directions for QDP. By summarizing recent advancements, we hope to provide a comprehensive, up-to-date review for researchers venturing into this field.