Federated Unlearning of Quantum Data via Classical Shadows

Technical Summary: Federated Unlearning of Quantum Data via Classical Shadows

Problem Statement
Quantum Federated Learning (QFL) enables the training of Quantum Neural Networks (QNNs) across distributed nodes holding quantum data, such as sensor states, without centralizing the data. However, QFL systems face a critical challenge: the "right to be forgotten." In classical federated learning, unlearning often involves retraining or replaying stored data. In the quantum domain, this is fundamentally impossible due to the no-cloning theorem and the fact that quantum states are consumed upon measurement. Furthermore, existing federated unlearning methods that rely on stored gradient updates can only remove entire nodes, failing to handle granular requests such as removing a specific time window of data (e.g., from a drifting sensor) or a subset of samples. The core problem is how to remove the influence of specific quantum states from a trained QNN without access to the original states or the ability to re-measure them.

Methodology: Shadow-based Data Erasure (SHADE)
The authors propose SHADE, a framework that enables unlearning in QFL by leveraging classical shadows. The methodology operates in three phases:

1. Shadow Recording (Training Phase):
   During training, each node splits the available copies of every quantum state. While some copies are used for standard parameter-shift training, a subset is reserved for recording "classical shadows." Instead of storing the full quantum state (which is impossible), the node performs random single-qubit Pauli measurements on these copies. Crucially, the node only stores the measurement outcomes and basis choices for qubits within the backward light cone of the observable used in the QNN. This ensures that the storage cost scales with the circuit depth (light cone size) rather than the total number of qubits. These snapshots are kept locally and never uploaded.

2. Estimation (Unlearning Request Phase):
   When a forget request arrives (specifying a set of states $D_f$), the server broadcasts the current model parameters $\theta^*$. Nodes use their stored shadows to estimate the gradient and Hessian (curvature) of the loss function with respect to the forget data and the retained data at the current parameters $\theta^*$. Because classical shadows allow the prediction of expectation values for observables chosen after measurement, nodes can compute these estimates without re-preparing the quantum states.

3. Federated Newton Correction:
   The server aggregates the gradient and Hessian estimates from all nodes. It then applies a single Newton step to the model parameters. This correction effectively "undoes" the pull of the forget data along the curvature of the retained data, approximating the model that would have been obtained if the forget data had never been seen. Finally, the nodes erase the specific shadows associated with the forget data to ensure the information is permanently removed.

Key Contributions

• Formulation of Quantum Unlearning: The paper formulates the unlearning problem in QFL where re-measurement is impossible, framing it as an optimization problem to minimize the number of recorded copies ($S$) while meeting a target error bound.
• SHADE Algorithm: The proposal of a practical algorithm that records classical shadows on the backward light cone during training and utilizes them for a federated Newton correction to remove data influence.
• Theoretical Guarantees:
  • Unbiased Estimation: Proof that shadow estimates for gradients and curvatures are unbiased at any model parameters.
  • Complexity Bounds: Proof that the number of snapshots ($S$) required to achieve a target unlearning error grows exponentially with the size of the backward light cone (circuit depth) but not with the total number of qubits. This makes the approach scalable for shallow circuits.
  • Error Bounds: Derivation of bounds on the distance between the unlearned model and the ideal retrained model.
• Empirical Validation: Demonstration of the method in simulation (50 nodes, 8 qubits) and on edge hardware (Jetson boards with simulated IBM noise), showing performance comparable to retraining.

Experimental Results
The authors evaluated SHADE against baselines including FedRecovery (node removal only), QMU, SCRUB, and CR (which require fresh data copies), as well as a "Retrain" baseline.

• Accuracy: SHADE achieves forget accuracy (FA) within 0.5 percentage points of the retrained model across node, window, and sample deletion scenarios.
• Efficiency: The correction is completed in under 0.2 seconds, significantly faster than baselines like CR (which takes thousands of seconds).
• Resource Usage: SHADE requires no replay of quantum states. The storage cost is approximately 72 kB per state (for specific parameters), and uplink communication is comparable to standard gradient uploads.
• Scalability: The number of required snapshots remains of the same order for $n=8$ and $n=12$ qubits (provided the circuit depth/light cone remains constant), and the correction time remains approximately constant (around 0.2 s) across the tested range of node counts.

Significance and Claims
The paper claims that SHADE provides the first viable solution for granular unlearning in quantum federated learning where data cannot be replayed. By exploiting the locality of quantum observables (the backward light cone), the method circumvents the exponential storage costs typically associated with quantum state tomography. The authors assert that their approach allows QFL systems to comply with data deletion requests (e.g., for drifting sensors or withdrawn consent) without compromising the utility of the retained data or requiring the impossible re-preparation of quantum states. The results suggest that with a modest classical storage overhead, QFL networks can maintain the "right to be forgotten" with accuracy nearly indistinguishable from full retraining.