Abstract We investigate the optimization of graph topologies for quantum sensing networks designed to estimate weak magnetic fields. The sensors are modeled as spin systems governed by a transverse-field Ising Hamiltonian in thermal equilibrium at low temperatures. Using a genetic algorithm (GA), we evolve network topologies to maximize a perturbative spectral sensitivity measure, which serves as the fitness function for the GA. For the best-performing graphs, we compute the corresponding quantum Fisher information (QFI) to assess the ultimate bounds on estimation precision. To enable efficient scaling, we use the GA-generated data to train a deep neural network, allowing extrapolation to larger graph sizes where direct computation becomes prohibitive. Our results show that while both the fitness function and QFI initially increase with system size, the QFI exhibits a clear non-monotonic behavior—saturating and eventually declining beyond a critical graph size. This reflects the loss of superlinear scaling of the QFI, as the narrowing of the energy gap signals a crossover to classical scaling of the QFI with system size. The effect is reminiscent of the microeconomic law of diminishing returns: beyond an optimal graph size, further increases yield reduced sensing performance. This saturation and decline in precision are particularly pronounced under Kac scaling, where both the QFI and spin squeezing plateau or degrade with increasing system size. We also attribute observed even–odd oscillations in the spectral sensitivity measure and QFI to quantum interference effects in spin phase space, as confirmed by our phase-space analysis. These findings highlight the critical role of optimizing interaction topology—rather than simply increasing network size—and demonstrate the potential of hybrid evolutionary and learning-based approaches for designing high-performance quantum sensors.
Abstract We investigate the optimization of graph topologies for quantum sensing networks designed to estimate weak magnetic fields. The sensors are modeled as spin systems governed by a transverse-field Ising Hamiltonian in thermal equilibrium at low temperatures. Using a genetic algorithm (GA), we evolve network topologies to maximize a perturbative spectral sensitivity measure, which serves as the fitness function for the GA. For the best-performing graphs, we compute the corresponding quantum Fisher information (QFI) to assess the ultimate bounds on estimation precision. To enable efficient scaling, we use the GA-generated data to train a deep neural network, allowing extrapolation to larger graph sizes where direct computation becomes prohibitive. Our results show that while both the fitness function and QFI initially increase with system size, the QFI exhibits a clear non-monotonic behavior—saturating and eventually declining beyond a critical graph size. This reflects the loss of superlinear scaling of the QFI, as the narrowing of the energy gap signals a crossover to classical scaling of the QFI with system size. The effect is reminiscent of the microeconomic law of diminishing returns: beyond an optimal graph size, further increases yield reduced sensing performance. This saturation and decline in precision are particularly pronounced under Kac scaling, where both the QFI and spin squeezing plateau or degrade with increasing system size. We also attribute observed even–odd oscillations in the spectral sensitivity measure and QFI to quantum interference effects in spin phase space, as confirmed by our phase-space analysis. These findings highlight the critical role of optimizing interaction topology—rather than simply increasing network size—and demonstrate the potential of hybrid evolutionary and learning-based approaches for designing high-performance quantum sensors.
We introduce Genetic Transformer Assisted Quantum Neural Networks (GTQNNs), a hybrid learning framework that combines a transformer encoder with a shallow variational quantum circuit and automatically fine tunes the circuit via the NSGA-II multi objective genetic algorithm. The transformer reduces high-dimensional classical data to a compact, qubit sized representation, while NSGA-II searches for Pareto optimal circuits that (i) maximize classification accuracy and (ii) minimize primitive gate count an essential constraint for noisy intermediate-scale quantum (NISQ) hardware. Experiments on four benchmarks (Iris, Breast Cancer, MNIST, and Heart Disease) show that GTQNNs match or exceed state of the art quantum models while requiring much fewer gates for most cases. A hybrid Fisher information analysis further reveals that the trained networks operate far from barren plateaus; the leading curvature directions increasingly align with the quantum subspace as the qubit budget grows, confirming that the transformer front end has effectively condensed the data. Together, these results demonstrate that GTQNNs deliver competitive performance with a quantum resource budget well suited to present-day NISQ devices.
The precision of a quantum thermometer is fundamentally constrained by the spectral structure of the probe itself, and a systematic mapping between the configurations of energy levels and thermometric performance provides relevant information to design optimized devices. In this work, we establish such a mapping by analyzing a broad class of quantum systems, ranging from finite spin ensembles and degenerate atoms to confining potentials, quantum walks, and continuous-spectrum models. We derive exact scaling laws for the quantum Fisher information, revealing two distinct high-temperature universality classes: finite-spectrum probes exhibit a T−4 decay, while unbounded or continuous spectra yield a slower T−2 decay. At low temperatures, we show that sensitivity, though universally exponentially suppressed, can be enhanced arbitrarily by engineering degenerate excited states or a quantum walk on a fully connected topology. By contrast, specific quantum walk topologies provide a distinct enhancement mechanism based on gap engineering, whereby an optimal network size yields an optimized T−2 low-temperature scaling. Furthermore, power-law spectra enable tunable scaling of thermometric performance with system size, offering a design principle for optimal probes in specific temperature windows. Our results contribute to transform spectral information into a resource for quantum thermometry, providing both fundamental bounds and practical guidelines to tailored temperature sensing.
Quantum systems used for metrology can offer enhanced precision over their classical counterparts. The design of quantum sensors can be optimized by maximizing the quantum Fisher information (QFI), which characterizes the precision of parameter estimation for an ideal measurement. Here we consider the response of a quantum system as a means to estimate the strength of a weak external perturbation. General expressions for the QFI in the nonequilibrium steady-state are derived, which hold for arbitrary interacting many-body systems at finite or zero temperature, and can be related to susceptibilities or linear-response transport coefficients. For quantum dot nanoelectronics devices, we show that electron interactions can lead to *exponential* scaling of the QFI with system size, highlighting that quantum resources can be utilized in the full Fock space. The precision estimation of voltages and fields can also be achieved by practical measurements. In particular, we show that current-based metrology in quantum circuits can leverage many-body effects for enhanced sensing.
Optical quantum sensing promises measurement precision beyond classical sensors termed the Heisenberg limit (HL). However, conventional methodologies often rely on prior knowledge of the target system to achieve HL, presenting challenges in practical applications. Addressing this limitation, we introduce an innovative deep-learning-based quantum sensing scheme (DQS), enabling optical quantum sensors to attain HL in agnostic environments. DQS incorporates two essential components: a graph neural network (GNN) predictor and a trigonometric interpolation algorithm. Operating within a data-driven paradigm, DQS utilizes the GNN predictor, trained on offline data, to unveil the intrinsic relationships between the optical setups employed in preparing the probe state and the resulting quantum Fisher information (QFI) after interaction with the agnostic environment. This distilled knowledge facilitates the identification of optimal optical setups associated with maximal QFI. Subsequently, DQS employs a trigonometric interpolation algorithm to recover the unknown parameter estimates for the identified optical setups. Extensive experiments are conducted to investigate the performance of DQS under different settings up to eight photons. Our findings not only offer a different lens through which to accelerate optical quantum sensing tasks but also catalyze future research integrating deep learning and quantum mechanics. Published by the American Physical Society 2024
Quantum Embeddings (QE) is an important component of Quantum Machine Learning (QML) algorithms to load classical data present in Euclidean space onto quantum Hilbert space, which are then later forwarded to the Parametric Quantum Circuit (PQC) for training and finally measured to compute the cost classically. The performance of the QML algorithm can vary according to the type of QE used, and also on mapping of features within the embedding (i.e., onto the qubits). This provides motivation to search for the optimal quantum embedding (i.e., feature to qubit mapping). Typically this problem is presented as an optimization problem, where the quantum embedding has trainable parameters and the optimal embedding is found out by training the embedding. In this work, we present identification of the optimal embedding as a search problem rather than an optimization problem. We show that for fixed number of qubits and model initialization of a Quantum Neural Network (QNN), different mapping of features onto qubits via QE changes the final performance of the QML algorithm. We propose Genetic Algorithm (GA) based search to find the optimal mapping of the features to the qubits. We perform experiments to find the optimal QE for binary classes of MNIST and Tiny Imagenet datasets and compare the results with randomly selected feature to qubit mapping (under identical number of runs). Our results show that GA-based approach performs better than random selection for MNIST (Tiny Imagenet) by 0.33-3.33 (0.5-3.36) higher average fitness score with up to 8.1%-15% (for MNIST) and 5.3%-8.8% (for Tiny Imagenet) less runtime. For both the datasets increasing the qubit counts marginally affects the GA fitness implying that the GA is scalable both in terms of dataset and in terms of QNN size. Compared to existing methodologies such as Quantum Embedding Kernel (QEK), Quantum Approximate Optimization Algorithm (QAOA)-based embedding and Quantum Random Access Code (QRAC), GA performs better by 1.003X, 1.03X and 1.06X respectively.
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.