QML for Quantum Sensing under Measurement-Induced Information Loss

Technical Summary: QML for Quantum Sensing under Measurement-Induced Information Loss

Problem Statement
Nitrogen-vacancy (NV) centers in diamond are promising solid-state quantum sensors for high-sensitivity magnetometry. However, in the noisy intermediate-scale quantum (NISQ) era, extracting reliable information from sensing data is challenging due to noise, decoherence, and the limitations of finite-shot measurements. A critical bottleneck in current workflows is measurement-induced information loss. In standard protocols, the physical signal (e.g., a magnetic field) is encoded into the coherent quantum state of the sensor. However, practical experiments only provide finite measurement statistics obtained after projective readout, which irreversibly discards phase information and obscures the underlying quantum dynamics.

While Quantum Machine Learning (QML) is often proposed to improve parameter estimation by leveraging high-dimensional quantum feature spaces, it remains unclear whether performance gains stem from the expressive power of the learning model or from access to coherent quantum information before measurement. This work investigates whether QML offers an advantage when restricted to post-measurement classical data versus when it has access to pre-measurement coherent quantum states.

Methodology
The authors formulated magnetic-field estimation as a supervised regression task within an NV center-inspired simulation. The experimental pipeline involved:

1. Sensing Protocol: A Ramsey interferometry sequence was simulated where an unknown magnetic field $B$ induces a phase shift $\phi = \gamma B t$ on a qubit initialized in a superposition state.
2. Noise Modeling: Realistic constraints were applied, including a phase-damping channel ($T_2 = 100 \mu s$), 10% classical bit-flip readout error, and shot noise modeled via binomial sampling.
3. Data Generation: For each magnetic field value, the system evolved at multiple time points ($t_k \in \{5, 10, 20, 40, 60\} \mu s$).
4. Learning Pipelines: Two distinct pipelines were constructed to isolate the impact of data availability:
   • Classical Pipeline: Models were trained on post-measurement statistics. Features included expectation values of Pauli operators ($\langle Z \rangle$ only, or $\langle X \rangle, \langle Y \rangle, \langle Z \rangle$) derived from finite shots. Models tested included Linear Ridge Regression (LRR), Radial Basis Function Kernel Ridge Regression (RBF-KRR), and Multilayer Perceptrons (MLP).
   • Quantum Pipeline: Models were trained on pre-measurement coherent quantum states represented as density matrices ($\rho$). This pipeline utilized a Quantum Kernel Ridge Regression (Q-KRR) model with a fidelity-based kernel (Uhlmann fidelity) computed directly from the density matrices.
5. Comparative Benchmarks:
   • Oracle Quantum Kernel: Trained on exact, noise-free pre-measurement density matrices to establish a theoretical upper bound on performance.
   • Tomography-Limited Quantum Kernel: Trained on density matrices reconstructed from measured Pauli expectation values, simulating a realistic scenario where quantum data is inferred from classical measurements.

Key Contributions

1. Benchmark Formulation: The authors established a regression benchmark for NV center-inspired magnetometry that explicitly separates the effects of model expressivity from data accessibility under realistic measurement constraints.
2. Quantification of Information Loss: By comparing classical models trained on measurement statistics against quantum models trained on coherent states, the work quantifies the performance gap imposed by measurement-induced information loss.
3. Theoretical Upper Bound: The study establishes a theoretical upper bound for sensing performance using quantum kernel ridge regression on pre-measurement density matrices, demonstrating the maximum potential accuracy achievable if coherent information is preserved.

Results

• Performance Gap: The oracle quantum kernel (trained on exact pre-measurement density matrices) achieved a near-constant, significantly lower error floor ($\sim 7 \times 10^{-4} \mu T$) compared to all measurement-based models.
• Classical Limitations: Classical models (both linear and nonlinear) trained on post-measurement statistics, even with rich feature sets (XYZ) and increased shot counts, deviated significantly from the theoretical upper bound. The paper notes that the estimation error in the measurement-based models is "100 times more" than that of the theoretical upper bound, indicating a substantial performance deficit.
• Model Complexity vs. Data Access: Increasing the complexity of classical models (e.g., moving from linear to MLP) improved performance but did not close the gap with the quantum oracle. Conversely, the tomography-limited quantum kernel (which relies on reconstructed states) performed similarly to strong classical models, indicating that the advantage is lost when coherent information is not directly accessible.
• Primary Driver: The results indicate that the primary source of performance improvement in QML-based sensing is the access to coherent quantum-state information rather than the intrinsic expressivity of the learning model architecture.

Significance and Claims
The paper concludes that the potential advantage of QML in quantum sensing is not solely a function of designing more complex learning models. Instead, it is strongly linked to the preservation and direct access of quantum coherence before it is collapsed into classical measurement outcomes.

The authors argue that current sensing workflows, which convert quantum states to classical statistics before inference, fundamentally constrain learning performance. To realize the full potential of QML in quantum sensing, future architectures must integrate sensing, state processing, and learning more closely within the quantum domain. This "quantum-native" approach is necessary to minimize information loss and enhance parameter estimation under realistic constraints. The work does not claim that QML will solve all sensing problems immediately but highlights that without preserving coherent data, the theoretical benefits of quantum learning remain inaccessible.