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Journal of Electronic Imaging

Noisy normal arbitrary quantum superposition state reconstruction based on parameterized quantum circuits

Hai-Sheng Li, Yuxing Wei, Cong Hu

Published Jun 4, 2025• 0 citations
Publisher

Abstract

To efficiently reconstruct noisy quantum states, specifically the normal arbitrary superposition state (NASS), on a quantum computer, we propose a reconstruction scheme based on parameterized quantum circuits (PQCs). First, the NASS image representation method is adopted, encoding images as quantum states, and a PQC is designed for reconstructing NASS. Then, gradient descent is used to train and update the circuit parameters. Finally, experiments validate the reconstruction of the quantum state. Experimental results show that for noisy grayscale image reconstruction, the structural similarity index measure reaches 0.922, and the peak signal-to-noise ratio is 26.74 dB. In terms of algorithmic complexity, the proposed scheme has a complexity of 4n2−4n+1, requiring no ancillary qubits. The proposed reconstruction scheme for noisy NASS quantum states offers the advantages of requiring fewer qubits and lower complexity while achieving effective reconstruction performance.

DOI
10.1117/1.jei.34.3.033042

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To efficiently reconstruct noisy quantum states, specifically the normal arbitrary superposition state (NASS), on a quantum computer, we propose a reconstruction scheme based on parameterized quantum circuits (PQCs). First, the NASS image representation method is adopted, encoding images as quantum states, and a PQC is designed for reconstructing NASS. Then, gradient descent is used to train and update the circuit parameters. Finally, experiments validate the reconstruction of the quantum state. Experimental results show that for noisy grayscale image reconstruction, the structural similarity index measure reaches 0.922, and the peak signal-to-noise ratio is 26.74 dB. In terms of algorithmic complexity, the proposed scheme has a complexity of 4n2−4n+1, requiring no ancillary qubits. The proposed reconstruction scheme for noisy NASS quantum states offers the advantages of requiring fewer qubits and lower complexity while achieving effective reconstruction performance.
In order to gain comprehensive knowledge of an arbitrary unknown quantum state, one feasible way is to reconstruct it, which can be realized by finding a series of quantum operations that can refactor the unitary evolution producing the unknown state. We design an adaptive framework that can reconstruct unknown quantum states at high fidelities, which utilizes SWAP test, parameterized quantum circuits (PQCs) and layerwise learning strategy. We conduct benchmarking on the framework using numerical simulations and reproduce states of up to six qubits at more than 96% overlaps with original states on average using PQCs trained by our framework, revealing its high applicability to quantum systems of different scales theoretically. Moreover, we perform experiments on a five-qubit IBM Quantum hardware to reconstruct random unknown single qubit states, illustrating the practical performance of our framework. For a certain reconstructing fidelity, our method can effectively construct a PQC of suitable length, avoiding barren plateaus of shadow circuits and overuse of quantum resources by deep circuits, which is of much significance when the scale of the target state is large and there is no a priori information on it. This advantage indicates that it can learn credible information of unknown states with limited quantum resources, giving a boost to quantum algorithms based on parameterized circuits on near-term quantum processors.
Parameterized Quantum Circuits (PQC) are promising towards quantum advantage on near-term quantum hardware. However, due to the large quantum noises (errors), the performance of PQC models has a severe degradation on real quantum devices. Take Quantum Neural Network (QNN) as an example, the accuracy gap between noise-free simulation and noisy results on IBMQ-Yorktown for MNIST-4 classification is over 60%. Existing noise mitigation methods are general ones without leveraging unique characteristics of PQC; on the other hand, existing PQC work does not consider noise effect. To this end, we present QuantumNAT, a PQC-specific framework to perform noise-aware optimizations in both training and inference stages to improve robustness. We experimentally observe that the effect of quantum noise to PQC measurement outcome is a linear map from noise-free outcome with a scaling and a shift factor. Motivated by that, we propose post-measurement normalization to mitigate the feature distribution differences between noise-free and noisy scenarios. Furthermore, to improve the robustness against noise, we propose noise injection to the training process by inserting quantum error gates to PQC according to realistic noise models of quantum hardware. Finally, post-measurement quantization is introduced to quantize the measurement outcomes to discrete values, achieving the denoising effect. Extensive experiments on 8 classification tasks using 6 quantum devices demonstrate that QuantumNAT improves accuracy by up to 43%, and achieves over 94% 2-class, 80% 4-class, and 34% 10-class classification accuracy measured on real quantum computers. The code for construction and noise-aware training of PQC is available in the TorchQuantum library.
Parameterized quantum circuits (PQCs) are fundamental to many hybrid quantum-classical algorithms. However, existing structure-based optimizers for the training of PQCs, such as Rotosolve and sequential minimal optimization, rely on heuristic node selection or ignore statistical noise, which limits their robustness and accuracy. To address this issue, we propose an interpolation-based coordinate descent (ICD) method as a unified framework for all structure-based optimizers. ICD approximates the cost function through interpolation, recovers its trigonometric structure, and performs global one-dimensional updates on individual parameters. Unlike previous methods, ICD derives optimal interpolation nodes that minimize statistical errors from measurements. For the common case of r equidistant frequencies, we prove that equidistant nodes with spacing 2π/(2r + 1) jointly minimize the mean squared error of Fourier coefficient estimates, the condition number of the interpolation matrix, and the average variance of the approximated cost function. Numerical experiments confirm the superior robustness and efficiency of ICD over gradient-based methods. Parameterized quantum circuits are a common tool in variational quantum algorithms and quantum machine learning. The authors design an interpolation-based coordinate descent method that reconstructs the cost landscape from a few circuit runs and achieves more efficient training than standard gradient and coordinate descent methods in our numerical tests.
The classical-quantum interface for loading classical data into quantum systems is an indispensable component of quantum information processing, and the parameterized quantum circuit (PQC) represents significant methodology in this area. However, a challenge in designing parameterized quantum circuits that can fully utilize limited qubit resources while accurately representing classical data remains. To address this issue, we propose the enhanced natural parameterized quantum circuit (ENPQC), which achieves the maximum parameter capacity of quantum systems and preserves the local structure of the original dataset. To make the encoding circuits more experiment friendly, we further provide two near-optimal circuits that achieve near-maximum capacity and are comparable in complexity to the existing PQCs. We numerically show that the ENPQC overwhelms other encoding methods for multiple datasets, demonstrating its potential for machine learning tasks. Furthermore, we experimentally validate our scheme by demonstrating a three-class classification task on an NMR platform, which achieves over 97 % accuracy. This work provides a powerful tool for the classical-quantum interface, paving the way for quantum big data processing during the noisy intermediate-scale quantum era.
Noisy intermediate-scale quantum computers are useful for various tasks such as state preparation and variational quantum algorithms. However, the non-Euclidean quantum geometry of parameterized quantum circuits is detrimental for these applications. Here we introduce the natural parameterized quantum circuit (NPQC) that can be initialized with a Euclidean quantum geometry. The initial training of variational quantum algorithms is substantially sped up as the gradient is equivalent to the quantum natural gradient. Further, we show how to estimate the parameters of the NPQC by sampling the circuit, which could be used for benchmarking or calibrating NISQ hardware. For a general class of quantum circuits, the NPQC has the minimal quantum Cram\'er-Rao bound, which highlights its potential for quantum metrology. Finally, we show how to generate arbitrary superpositions of two states with the NPQCs for state preparation tasks. Our results can be used to enhance currently available quantum processors.
Image classification is a core task in the field of computer vision, with significant implications for applications such as medical imaging and autonomous driving. Classical neural network architectures face numerous challenges when dealing with high-dimensional and complex data, particularly in terms of feature extraction and data processing capabilities, limiting their performance in certain applications. Quantum computing offers promising new approaches to processing high-dimensional complex data, most notably through Parameterized Quantum Circuits (PQC) as quantum hybrid neural networks for image classification. In this paper, we propose a hybrid neural network featuring a specific PQC structure designed to generate entanglement in accordance with sample images. This PQC structure utilizes the adjacency matrix of the corresponding graph of the sample image, embedding adjacency relationships among nodes into the entanglement module, thereby generating entanglement aligned with the features of specific images. Numerical results validate the effectiveness of our PQC model across various image classification tasks, enhancing the efficiency of feature learning and representation. Our model demonstrates significant advantages in classification accuracy and learning efficiency on classical datasets. Our research not only advances the theoretical understanding of hybrid quantum networks but also paves the way for practical implementations in real-world scenarios.