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QML for Quantum Sensing under Measurement-Induced Information Loss

Sounak Bhowmik, Himanshu Thapliyal

arXiv:2608.23934v1quant-phcs.AI

TL;DR

The paper asks whether QML sensing gains come from model expressivity or from preserving coherent information that measurement can destroy. It compares classical regression on post-measurement statistics with quantum-kernel regression on pre-measurement density matrices in an NV-inspired magnetometry simulation, finding much lower error with coherent-state access and identifying closer sensor–learning integration as a future direction.

  • Problem

    Measurement noise, decoherence, shot noise, and finite sampling can obscure magnetic-field information before learning, leaving unclear whether QML gains arise from model expressivity or coherent-state access.

  • Method

    The study formulates NV-inspired magnetic-field sensing as supervised regression and compares classical models on finite measurement statistics with quantum-kernel models on pre-measurement density matrices.

  • Results

    Quantum models using coherent quantum-state representations achieve much lower estimation error and define a theoretical upper bound, whereas lossy measurements constrain models trained on classical data.

  • Takeaways & Limitations

    The potential QML advantage is linked strongly to preserving and using quantum coherence, motivating sensing architectures that integrate state processing and learning more closely within the quantum domain.

  • Takeaways & Limitations

    The study remains a simulation-based analysis whose future extensions include hardware experiments, more complex noise models, and hybrid architectures under practical device limitations.

Abstract

from arXiv · show

Nitrogen-vacancy (NV) centers in diamond can serve as highly sensitive solid-state quantum sensors for high-sensitivity magnetometry. However, in the noisy intermediate-scale quantum (NISQ) era, extracting reliable information from noisy, finite-shot, and measurement-limited sensing data remains a considerable challenge. Whereas, quantum machine learning (QML) offers a potential path to improve parameter estimation by learning nonlinear relationships between quantum-sensing data and the underlying physical signal. In this work, we investigate the role of QML in magnetic-field estimation within an NV center-inspired magnetometry setting. We formulated magnetic field sensing as a supervised regression task. We compared the performance of several classical machine learning models trained on measurement-based classical data with that of quantum kernel-based models trained on pre-measurement coherent quantum states. Our objective is to isolate the impact of measurement-induced information loss and therefore provide a theoretical upper bound on the sensing performance. The upper bound is achievable only when coherent quantum information is directly available to the learning model. Our results show that QML-based sensing performance improves significantly with coherent quantum-state information, and not much with changes in model complexity or learning paradigm. This observation underscores the importance of learning pipelines that tightly integrate quantum sensors and QML models to enhance magnetic field sensing under realistic constraints.

I. INTRODUCTION

NV-center sensing combines coherent quantum evolution with practical measurement constraints that can discard field information before learning. This work compares classical models using measured statistics with quantum-kernel models using pre-measurement states to isolate that loss.

  • NV centers provide solid-state magnetometry with atomic-scale resolution, optical initialization and readout, microwave control, and room-temperature operation.
  • Finite measurement statistics, noise, decoherence, shot noise, and limited sampling can remove or obscure magnetic-field information before it reaches a learning algorithm.
  • The study asks whether sensing improvements arise mainly from model expressivity or from access to coherent quantum information before measurement.
  • The authors formulate magnetic-field estimation as supervised regression and compare classical models trained on measurement statistics with quantum kernel ridge regression trained on coherent density matrices.
  • The reported results show substantially better estimation with coherent density matrices, while measured-data models remain constrained by readout quality and completeness.
  • Quantum-kernel regression on pre-measurement density matrices supplies a theoretical upper bound on sensing performance before measurement-induced information loss.

II. BACKGROUND

NV centers are diamond point defects whose spin levels respond to magnetic fields and whose fluorescence enables optical initialization and readout. The sensing setup uses a two-level qubit representation for Ramsey interferometry.

  • NV centers are atomic-scale point defects formed by a nitrogen atom adjacent to a vacant carbon lattice site.
  • Zero-field splitting separates the |0⟩ state from degenerate |+1⟩ and |−1⟩ states, while an axial magnetic field produces proportional Zeeman splitting.
  • Spin-dependent fluorescence enables optical initialization and readout, and microwave excitation drives transitions between |0⟩ and |±1⟩.
  • NV magnetometry experiments are restricted here to the two-level system formed by ms = {0, −1}, suitable for Ramsey interferometry.

B. NV Inspired Quantum Sensing Simulation

The simulation isolates Zeeman-driven field dynamics in a two-level NV-inspired system using phase-accumulating evolution and Ramsey pulses. Measurement converts encoded quantum-state information into statistics while discarding phase information.

  • The effective Hamiltonian isolates the field-dependent Zeeman interaction as H = ℏγBSz, with B as the unknown magnetic field.
  • The two-level circuit represents evolution with a Z-axis rotation whose angle depends on magnetic field and time.
  • A π/2 pulse creates a sensitive superposition state before evolution under the field-dependent unitary.
  • The pipeline estimates the unknown field from either coherent density matrices or classical expectation values after sensing evolution.
  • A second Hadamard gate returns the evolved state to the computational basis before measurement.
  • Z-basis measurement irreversibly discards phase information encoded in density-matrix off-diagonal elements, producing data loss.
  • The Ramsey phase is ϕ = γBtk, where tk denotes the evolution time.

III. EXPERIMENTAL SETUP

The experimental setup compares classical regression on measured expectation values with quantum-kernel regression on stored pre-measurement density matrices. A fixed phase-damping, readout-error, and shot-noise model makes the comparison measurement-limited.

  • The controlled experiment prepares superposition states, evolves them under an unknown magnetic field, and uses the resulting data to train regression models.
  • Classical models use expectation-value features {⟨Z⟩t1, ⟨Z⟩t2, ..., ⟨Z⟩tM}, whereas the quantum model uses coherent-state sets Ri = {ρi(t1), ρi(t2), ..., ρi(tM)}.
  • The noise model includes phase damping with T2 = 100 µs, 10% classical bit-flip readout error, and binomial shot-noise sampling.
  • The system evolves at tk ∈ {5, 10, 20, 40, 60} µs, with noise and sensing parameters fixed throughout the experiments.
  • Density matrices are stored before measurement for QML, while post-measurement expectation values serve as classical regression data.

IV. LEARNING FROM CLASSICAL AND QUANTUM DATA

The study separates model expressivity from quantum-data accessibility by comparing classical models using measurement-derived features with quantum models using coherent-state information.

  • The classical pipeline receives measurement-derived features, whereas the quantum pipeline uses the same sensing dynamics with different information available to the learner.This comparison isolates information lost during measurement from limitations of model capacity.
  • The benchmark includes linear ridge regression, nonlinear RBF kernel ridge regression and MLP models, and quantum kernel ridge regression.Quantum models use oracle or tomography-reconstructed density matrices.

A. Classical Learning From Measurement Records

The classical pipeline learns magnetic fields from finite-shot measurement statistics rather than directly accessed quantum states. Comparing feature maps and model capacities helps isolate information loss caused by measurement from limitations of linear regression.

  • A. Classical Learning From Measurement Records: Each training instance samples a magnetic field, runs Ramsey sensing across evolution times, and converts measured states into a finite-shot feature vector.The learner receives measurement statistics, not the underlying quantum state.
  • A. Classical Learning From Measurement Records: The Z-only feature map uses computational-basis readout, which loses phase information encoding the magnetic field during Ramsey evolution.This is a restricted measurement protocol with direct information loss.
  • A. Classical Learning From Measurement Records: The XYZ feature map measures three Pauli bases and partially captures coherence, but finite-shot sampling and readout error remain limiting factors.Quantum-state information must still be estimated from classical measurement outcomes.
  • A. Classical Learning From Measurement Records: Ridge regression provides the linear baseline, while RBF-kernel ridge regression and a multilayer perceptron test whether nonlinear model capacity changes performance.The comparison separates measurement-induced information loss from limited linear-model capacity.

B. Quantum Learning From Pre-Measurement Quantum States

The quantum pipeline applies kernel ridge regression to full density-matrix trajectories available before final measurement. This preserves coherent phase information and represents a theoretical upper bound whose gap from measurement-based models can be reduced through closer sensor-learning integration or richer measurement strategies.

  • B. Quantum Learning From Pre-Measurement Quantum States: The quantum pipeline assumes access to full density matrices generated across Ramsey evolution times for each sampled magnetic field.It uses the same physical sensing process as the classical pipeline but omits final-measurement access limitations.
  • B. Quantum Learning From Pre-Measurement Quantum States: The density-matrix trajectory preserves coherent phase information in the off-diagonal elements.This information is unavailable to learners receiving only measurement statistics.
  • B. Quantum Learning From Pre-Measurement Quantum States: Fidelity-based quantum kernel ridge regression compares pairs of sensing trajectories using the Uhlmann fidelity between density matrices.The kernel is evaluated over quantum-state trajectories rather than classical feature vectors.
  • B. Quantum Learning From Pre-Measurement Quantum States: The predicted magnetic field is obtained from the kernel Gram matrix and the training magnetic-field values.The prediction uses kernel ridge regression over the trajectory representation.
  • B. Quantum Learning From Pre-Measurement Quantum States: 100 times more estimation error occurs for models learning from measurement data than for the theoretical upper bound.The passage attributes closing this gap to tighter integration with coherent quantum states or more sophisticated measurement strategies.

C. Oracle and Tomography-Limited Quantum Kernels

The study compares oracle and tomography-limited quantum kernels to isolate how access to coherent quantum-state information affects learning performance under measurement constraints.

  • The oracle kernel uses exact pre-measurement density matrices to provide an upper bound on performance with direct coherent-state access.This setting is not intended to represent a directly available experimental measurement pipeline.
  • The tomography-limited kernel uses density matrices reconstructed from measured Pauli expectation values.
  • Tomography-based reconstruction remains constrained by finite-shot sampling, readout noise, and basis-measurement overhead.These constraints make it a more realistic comparison with classical learning methods.
  • Comparing classical, tomography-limited, and oracle kernels isolates the value of coherent quantum-state access as the central learning mechanism.

V. RESULTS

The results compare classical and quantum-kernel regression across finite-shot Ramsey measurements, showing that coherent-state access produces the strongest performance improvement.

  • 800 train and 100 test samples were used to evaluate magnetic-field regression performance.
  • Fields in [0.0, 2.0] µT were sampled, with Ramsey data generated under realistic dephasing noise at multiple evolution times.
  • The compared models included linear ridge regression, nonlinear RBF and MLP models, and quantum kernel ridge regression using oracle or reconstructed density matrices.
  • At 2048 shots, test RMSE was significantly lower for the theoretical upper bound than for measurement-based learning models.
  • Nonlinear classical models significantly outperform linear baselines across several regimes, particularly for Z-only features at higher shot counts.This indicates that part of the estimation problem is nonlinear in the measurement statistics.
  • The oracle quantum kernel reaches a near-constant error floor of ∼7×10^-4 µT, representing a theoretical upper bound without measurement loss.
  • The results indicate that sensing gains primarily stem from coherent quantum-state access rather than model expressivity alone.

VI. CONCLUSION

The study isolates measurement-induced information loss by comparing models trained on post-measurement statistics with quantum kernel models trained on coherent pre-measurement states. Results indicate that coherent quantum information, rather than model complexity alone, is central to improved sensing performance and motivates quantum-native sensing-learning pipelines.

  • The study formulates magnetic-field estimation as supervised regression and compares classical models using post-measurement statistics with quantum kernel models using coherent quantum states.This design isolates whether performance differences arise from the learning model or the information available in the data.
  • Lossy measurements fundamentally constrain models trained solely on measured classical data, whereas coherent quantum-state representations achieve much lower estimation error.The coherent-state models define a theoretical upper bound achievable when quantum information is directly accessible for learning.
  • The potential QML advantage is linked strongly to preserving and using quantum coherence, rather than solely to increased model complexity.
  • Future QML-enhanced sensing architectures should integrate sensing, state processing, and learning more closely within the quantum domain to reduce information loss.The study identifies hardware experiments, more complex noise models, and hybrid sensing-learning architectures as future directions.
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