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Distribution-Guided and Constrained Quantum Machine Unlearning

Nausherwan Malik, Zubair Khalid, Muhammad Faryad

TL;DR

This paper tackles the problem of removing the influence of forgotten data from variational quantum classifiers without full retraining. It introduces a distribution-guided forget target and an anchor-based preservation constraint, formulating unlearning as a constrained optimization with a Lagrangian interpretation and parameter-shift gradient optimization. Empirically, the method achieves sharp forgetting of the targeted class while maintaining retained-class performance and remains close to gold retrained baselines, demonstrated on Iris and Covertype with a low KL divergence ($\approx 0.047$) between the unlearned and retrained models on retained data. The work highlights the importance of data-driven target design and anchor constraints for reliable, interpretable quantum machine unlearning in near-term quantum devices.

Abstract

Machine unlearning aims to remove the influence of specific training data from a learned model without full retraining. While recent work has begun to explore unlearning in quantum machine learning, existing approaches largely rely on fixed, uniform target distributions and do not explicitly control the trade-off between forgetting and retained model behaviour. In this work, we propose a distribution-guided framework for class-level quantum machine unlearning that treats unlearning as a constrained optimization problem. Our method introduces a tunable target distribution derived from model similarity statistics, decoupling the suppression of forgotten-class confidence from assumptions about redistribution among retained classes. We further incorporate an anchor-based preservation constraint that explicitly maintains predictive behaviour on selected retained data, yielding a controlled optimization trajectory that limits deviation from the original model. We evaluate the approach on variational quantum classifiers trained on the Iris and Covertype datasets. Results demonstrate sharp suppression of forgotten-class confidence, minimal degradation of retained-class performance, and closer alignment with the gold retrained model baselines compared to uniform-target unlearning. These findings highlight the importance of target design and constraint-based formulations for reliable and interpretable quantum machine unlearning.

Distribution-Guided and Constrained Quantum Machine Unlearning

TL;DR

This paper tackles the problem of removing the influence of forgotten data from variational quantum classifiers without full retraining. It introduces a distribution-guided forget target and an anchor-based preservation constraint, formulating unlearning as a constrained optimization with a Lagrangian interpretation and parameter-shift gradient optimization. Empirically, the method achieves sharp forgetting of the targeted class while maintaining retained-class performance and remains close to gold retrained baselines, demonstrated on Iris and Covertype with a low KL divergence () between the unlearned and retrained models on retained data. The work highlights the importance of data-driven target design and anchor constraints for reliable, interpretable quantum machine unlearning in near-term quantum devices.

Abstract

Machine unlearning aims to remove the influence of specific training data from a learned model without full retraining. While recent work has begun to explore unlearning in quantum machine learning, existing approaches largely rely on fixed, uniform target distributions and do not explicitly control the trade-off between forgetting and retained model behaviour. In this work, we propose a distribution-guided framework for class-level quantum machine unlearning that treats unlearning as a constrained optimization problem. Our method introduces a tunable target distribution derived from model similarity statistics, decoupling the suppression of forgotten-class confidence from assumptions about redistribution among retained classes. We further incorporate an anchor-based preservation constraint that explicitly maintains predictive behaviour on selected retained data, yielding a controlled optimization trajectory that limits deviation from the original model. We evaluate the approach on variational quantum classifiers trained on the Iris and Covertype datasets. Results demonstrate sharp suppression of forgotten-class confidence, minimal degradation of retained-class performance, and closer alignment with the gold retrained model baselines compared to uniform-target unlearning. These findings highlight the importance of target design and constraint-based formulations for reliable and interpretable quantum machine unlearning.
Paper Structure (16 sections, 1 theorem, 13 equations, 5 figures, 2 tables)

This paper contains 16 sections, 1 theorem, 13 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Theoretical Foundations
  3. Formal Definition 1
  4. Formal Definition 2
  5. Quantum Machine Unlearning
  6. Methodology
  7. Distribution-Guided Class Unlearning
  8. Distribution-Guided Forget Target
  9. Anchor Reference Distributions
  10. Unlearning Objective
  11. Optimization and Interpretation
  12. Implementation Details
  13. Results
  14. Conclusion
  15. Limitations of uniform forget-target assignment
  16. ...and 1 more sections

Key Result

Theorem 1

Let $p_w(\cdot\mid x)$ denote the model softmax distribution induced by parameters $w$. Let $F$ be a forget set and $A$ be an anchor (retained) set. Let $p_{\mathrm{ref}}(\cdot\mid x) := p_{w_{\mathrm{orig}}}(\cdot\mid x)$ be the reference distribution produced by the original parameters $w_{\mathrm Then, maximizing the objective is equivalent, up to additive constants independent of $w$, to maxi

Figures (5)

  • Figure 1: Variational quantum classifier architecture. A four-feature data-encoding map is reuploaded twice and interleaved with a hardware-efficient ansatz on six qubits. Features are embedded via single-qubit rotations and pairwise entangling blocks, followed by repeated layers of parametrized $R_y$–$R_z$ rotations and ring CNOT entanglement. Pauli-$Z$ expectation values on the readout qubits are used to produce class logits.
  • Figure 2: Confusion matrix for the Iris dataset after training the variational quantum classifier. Rows correspond to true class labels and columns to predicted labels. The model achieves near-perfect separation across all three classes.
  • Figure 3: Confusion matrix for the Covertype dataset (classes 3, 5, and 7) after training. While correct predictions dominate the diagonal, residual confusion between classes reflects the increased complexity of the dataset compared to Iris.
  • Figure 4: Confusion matrices before (left) and after (right) targeted unlearning. (Left) Covertype dataset with class 2 forgotten. (Right) Iris dataset with class 2 forgotten. In both cases, predictions assigned to the forgotten class are strongly suppressed after unlearning, while errors primarily redistribute to a single retained class.
  • Figure 5: Effect of uniform forget-target assignment on the Covertype dataset. The confusion matrices show model predictions before (left) and after (center) unlearning using a uniform target distribution over non-forgotten classes, while the right panel reports the distribution of parameter changes. Uniform redistribution reduces, but does not eliminate, predictions assigned to the forgotten class, with the mean forgotten-class probability decreasing from $0.405$ to $0.373$. Probability mass is dispersed across retained classes rather than concentrated on a semantically similar class, resulting in less structured post-unlearning confusion patterns compared to similarity-guided unlearning.

Theorems & Definitions (1)

  • Theorem 1: Lagrangian formulation of distribution-guided unlearning