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Quantum-Enabled Probabilistic Optimal Power Flow with Built-in Differential Privacy

Yuji Cao, Tongxin Li, Yue Chen

Abstract

Quantum computing has been regarded as a promising approach to accelerate power system optimization. However, challenges such as limited qubits and inherent noise hinder their widespread adoption in power systems. In this paper, we propose a qubit-efficient framework for solving a crucial power system optimization problem, the probabilistic optimal power flow (POPF). We demonstrate that quantum noise, traditionally viewed as a drawback, can in fact be leveraged to provide a built-in differential privacy (DP) guarantee. Specifically, we first linearize POPF into a multi-parametric linear program (MP-LP) with renewable uncertainties being the parameters. This decomposes the parameter space into critical regions with precomputed solution maps. Second, a variational quantum circuit (VQC) classifies the critical region based on each uncertainty realization and then recovers the final solution. In this way, the required qubits scale with the uncertain parameters instead of the network size, with only 5 qubits versus 600+ for direct quantum OPF in a 69-bus system. Moreover, we prove the depolarizing noise of VQC provides DP guarantees and characterize the privacy-cost tradeoff. Case studies validate the proposed VQC achieves 2.1$\times$ smaller privacy budgets compared to its classical counterpart. At matched privacy levels, the VQC also maintains lower infeasibility and prediction error.

Quantum-Enabled Probabilistic Optimal Power Flow with Built-in Differential Privacy

Abstract

Quantum computing has been regarded as a promising approach to accelerate power system optimization. However, challenges such as limited qubits and inherent noise hinder their widespread adoption in power systems. In this paper, we propose a qubit-efficient framework for solving a crucial power system optimization problem, the probabilistic optimal power flow (POPF). We demonstrate that quantum noise, traditionally viewed as a drawback, can in fact be leveraged to provide a built-in differential privacy (DP) guarantee. Specifically, we first linearize POPF into a multi-parametric linear program (MP-LP) with renewable uncertainties being the parameters. This decomposes the parameter space into critical regions with precomputed solution maps. Second, a variational quantum circuit (VQC) classifies the critical region based on each uncertainty realization and then recovers the final solution. In this way, the required qubits scale with the uncertain parameters instead of the network size, with only 5 qubits versus 600+ for direct quantum OPF in a 69-bus system. Moreover, we prove the depolarizing noise of VQC provides DP guarantees and characterize the privacy-cost tradeoff. Case studies validate the proposed VQC achieves 2.1 smaller privacy budgets compared to its classical counterpart. At matched privacy levels, the VQC also maintains lower infeasibility and prediction error.
Paper Structure (20 sections, 2 theorems, 53 equations, 8 figures, 5 tables, 1 algorithm)

This paper contains 20 sections, 2 theorems, 53 equations, 8 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Probabilistic OPF and Linearization
  4. Multi-parametric Linear Programming Reformulation
  5. Hybrid Quantum-Classical Classifier for Probabilistic OPF
  6. Variational Quantum Feature Extractor
  7. Temperature-Controlled Region Classifier
  8. Privacy Analysis
  9. Inherent Quantum Noise
  10. Differential Privacy Guarantee
  11. Privacy-Cost Tradeoff
  12. Case Study
  13. Privacy Guarantees and Trade-offs
  14. Analysis of Operational Feasibility
  15. Qubit Efficiency and Computational Speedup
  16. ...and 5 more sections

Key Result

Theorem 1

Under Assumption ass:lipschitz, the randomized mechanism in Algorithm alg:pipeline that maps $\theta$ to the released region index $\tilde{k}$ satisfies $(\varepsilon_{\mathrm{reg}}, 0)$-differential privacy over the output space $\mathcal{Y} = \{1, \ldots, K\}$, where with $\|W_{cl}\|_{\infty,1} := \max_{k} \|w_k\|_1$ denoting the maximum $\ell_1$-norm of the rows of the weight matrix $W_{cl}$ i

Figures (8)

  • Figure 1: Variational quantum feature extractor with data reuploading technique. Each layer re-encodes input $\theta$ via $R_y$ gates, followed by trainable $U_\phi$ gates and CNOT entanglement. Measurements on all qubits yield the feature vector $h$.
  • Figure 2: Topology of 69-bus distribution network.
  • Figure 3: The critical regions of the multi-parametric LP problem for the 69-bus distribution network.
  • Figure 4: Privacy-utility frontier showing stochastic accuracy versus empirical $\varepsilon$ for different inverse temperatures $\beta$. Each curve traces the effect of varying noise $\gamma$ from 0 to 0.5.
  • Figure 5: Privacy-utility frontier comparison between VQC (with depolarizing noise) and multi-layer perceptron (MLP, with Gaussian logit noise). At matched privacy levels, the VQC maintains substantially higher accuracy.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Definition 1: Parameter Adjacency
  • Definition 2: Differential Privacy dworkRoth2014
  • Theorem 1: Region-ID Differential Privacy
  • Theorem 2: Privacy-Cost Tradeoff
  • Remark 1: Simplified Bound for Standard VQC
  • Remark 2: Tuning the Privacy-Cost Tradeoff