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arXiv:2503.20565 · Physical review. B./Physical review. B

Measuring incompatible observables with quantum neural networks

Muchun Yang, Yibin Huang, D. L. Zhou

Published Nov 8, 2025• 0 citations
arXivHTMLPDF

Abstract

The expressivity of quantum neural networks (QNNs) is a key issue in quantum machine learning (QML), which determines the capability to learn and represent complex functions. The recently proposed Randomness-Enhanced QNNs [Wu et al., Phys. Rev. Lett. 132, 010602 (2024)] use a mixed-unitary channel to accurately express arbitrary single observables. In this work, we generalize the mixed-unitary channel to a unital channel and prove that the unital channel can accurately express multiple arbitrary observables. We show that by training and implementing a unital quantum channel as a multiple-output QNN, we can accurately measure the expectation values of multiple incompatible observables simultaneously by Pauli-$Z$ measurements on distinct output qubits. We prove this accurate expressivity for multiple observables by computing the complete positivity of the unital quantum channel. An iterative algorithm is proposed to compute the theoretical maximal scaling factor of the QNNs' outcomes. Notably, our theoretical analysis shows that it requires fewer copies of state when measuring some incompatible observables by the multiple-output QNNs, which demonstrates a resource efficiency advantage over separately applying projective measurements.

DOI
10.1103/qz9c-m3z4

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As a hybrid of artificial intelligence and quantum computing, quantum neural networks (QNNs) have gained significant attention as a promising application on near-term, noisy intermediate-scale quantum devices. Conventional QNNs are described by parametrized quantum circuits, which perform unitary operations and measurements on quantum states. In this Letter, we propose a novel approach to enhance the expressivity of QNNs by incorporating randomness into quantum circuits. Specifically, we introduce a random layer, which contains single-qubit gates sampled from a trainable ensemble pooling. The prediction of QNN is then represented by an ensemble average over a classical function of measurement outcomes. We prove that our approach can accurately approximate arbitrary target operators using Uhlmann's theorem for majorization, which enables observable learning. Our proposal is demonstrated with extensive numerical experiments, including observable learning, Rényi entropy measurement, and image recognition. We find the expressivity of QNNs is enhanced by introducing randomness for multiple learning tasks, which could have broad application in quantum machine learning.
Quantum neural networks (QNNs), represented by parameterized quantum circuits, can be trained in the paradigm of supervised learning to map input data to predictions. Much work has focused on theoretically analyzing the expressive power of QNNs. However, in almost all literature, QNNs' expressive power is numerically validated using only simple univariate functions. We surprisingly discover that state-of-the-art QNNs with strong expressive power can have poor performance in approximating even just a simple sinusoidal function. To fill the gap, we propose four expressibility-enhancing strategies for QNNs: Sinusoidal-friendly embedding, redundant measurement, post-measurement function, and random training data. We analyze the effectiveness of these strategies via mathematical analysis and/or numerical studies including learning complex sinusoidal-based functions. Our results from comparative experiments validate that the four strategies can significantly increase the QNNs' performance in approximating complex multivariable functions and reduce the quantum circuit depth and qubits required.
Universality of neural networks describes the ability to approximate arbitrary function, and is a key ingredient to keep the method effective. The established models for universal quantum neural networks(QNN), however, require the preparation of multiple copies of the same quantum state to generate the nonlinearity, with the copy number increasing significantly for highly oscillating functions, resulting in a huge demand for a large-scale quantum processor. To address this problem, we propose a new QNN model that harbors universality without the need of multiple state-duplications, and is more likely to get implemented on near-term devices. To demonstrate the effectiveness, we compare our proposal with two popular QNN models in solving typical supervised learning problems. We find that our model requires significantly fewer qubits and it outperforms the other two in terms of accuracy and relative error.
Quantum neural networks (QNNs) have shown promise in leveraging quantum computation for machine learning tasks. Utilizing multiple identical photons as input, multi-photon quantum neural networks (MPQNNs) have the potential to enhance the expressivity through increasing the photon number. However, how precisely the expressivity of an MPQNN is affected by an increase in photon number, and whether it can be infinitely enhanced by increasing the photon number, remains unexplored. In this work, we quantitatively estimate the expressivity of this model by deriving upper bounds on approximation error in two cases. In the case of a fixed observable, there exists a threshold that scales linearly with the mode number. Below the threshold, the expressivity of an MPQNN can be enhanced polynomially by increasing the photon number. Above the threshold, however, increasing the photon number does not affect the expressivity. In the case of a trainable observable, the expressivity can always be enhanced polynomially by increasing the photon number. These findings are then validated by numerical simulations. Our work elucidates the performance enhancement of multi-photon quantum feature in QNNs, as well as its limitations, offering guidance for leveraging multi-photon advantages in quantum machine learning.
The field of quantum neural networks, an emerging interdisciplinary frontier bridging quantum computing and machine learning, has recently garnered substantial attention in the scientific community. In this work, we propose a complex-valued quantum neural network in the quantum circuit framework through expanding the number domains of conventional quantum machine learning architecture from the real-valued domain to the complex-valued plane. Such an expansion can alleviate the unitary constraints in quantum feature maps and optimization blocks, and introduce enhanced nonlinear characteristics. It is proven theoretically that the expressivity can be significantly enhanced through the frequency-dependent adaptability and the exponential or hyperbolic functional distribution extension for the Fourier coefficients of quantum models' outputs. Moreover, analysis based on covering numbers indicates that our algorithm can achieve an adaptive balance between expressivity and generalization via the trainable scaling factor, while exhibiting superior parameter efficiency relative to unitary quantum learning models. The performance of the proposed algorithms is demonstrated through extensive numerical simulations in regression and classification tasks, including an experimental realization on the superconducting quantum processor. Our work advances the harnessing of imaginarity resources of quantum theory for learning tasks, achieving a theoretically grounded performance enhancement, and underscores the potential of improving quantum neural networks through structured design for future applications.
Quantum neural networks (QNNs) require an efficient training algorithm to achieve practical quantum advantages. A promising approach is gradient-based optimization, where gradients are estimated by quantum measurements. However, QNNs currently lack general quantum algorithms for efficiently measuring gradients, which limits their scalability. To elucidate the fundamental limits and potentials of efficient gradient estimation, we rigorously prove a trade-off between gradient measurement efficiency (the mean number of simultaneously measurable gradient components) and expressivity in deep QNNs. This trade-off indicates that more expressive QNNs require higher measurement costs per parameter for gradient estimation, while reducing QNN expressivity to suit a given task can increase gradient measurement efficiency. We further propose a general QNN ansatz called the stabilizer-logical product ansatz (SLPA), which achieves the trade-off upper bound by exploiting the symmetric structure of the quantum circuit. Numerical experiments show that the SLPA drastically reduces the sample complexity needed for training while maintaining accuracy and trainability compared to well-designed circuits based on the parameter-shift method.