Variational Quantum Circuits Design Principles, Applications, and Challenges Toward Practical: A Review

Quantum Machine Learning (QML) is an emerging, powerful paradigm at the intersection of quantum computing and artificial intelligence, with the potential to enhance medical image analysis. We compared their performance with classical counterparts, while critically examining the gap between theoretical quantum advantages and the practical constraints of current Noisy Intermediate-Scale Quantum (NISQ) implementations. This review explored the current landscape of QML-based methodologies applied to key medical imaging modalities - Magnetic Resonance Imaging (MRI), Computed Tomography (CT), X-rays, and histopathology. We systematically analyzed how quantum-enhanced models were employed for critical tasks such as image classification, segmentation, and reconstruction, and we compared their performance with that of classical counterparts. The review delved into the architectural innovations that enabled these quantum approaches, including hybrid quantum-classical frameworks, variational quantum circuits, and quantum convolutional neural networks. Particular emphasis was given to the clinical relevance of QML applications, from early diagnostics to real-time surgery planning, and how quantum advantage might influence data efficiency, interpretability, along computational scalability. Finally, we highlighted current challenges in data encoding, noise resilience, and hardware limitations, and discussed promising future directions that could enable the practical deployment of QML in healthcare settings. This comprehensive review aimed to serve as a foundational reference for researchers and practitioners looking to harness quantum computing in medical imaging and precision medicine.

In recent years, Variational Quantum Circuit (VQC)s have been widely explored to advance quantum circuits against classic models on various domains, such as quantum chemistry and quantum machine learning. Similar to classic machine-learning models, VQCs can be optimized through gradient-based approaches. However, the gradient of VQCs may dramatically vanish during the optimization process as the number of qubits or layers increases. This issue, known as Barren Plateau (BP)s, seriously hinders the scaling of VQCs on large datasets. To mitigate the exponential gradient vanishing, extensive efforts have been devoted to tackling this issue through different methods. Some BP mitigation strategies include VQC parameter initialization and more complex VQC architectures such as Quantum Neural Network (QNN)s and Quantum Convolutional Neural Network (QCNN). In this thesis, we aim to investigate several circuit components through the lens of parameter initialization techniques. We study the performance and gradient variance of the different quantum circuit structures in combination with various initialization methods. In most existing studies, researchers typically focus on improving or understanding one aspect of the VQC approach. Instead, we plan to investigate how the two components of the VQC behave as we interchange the different variables of the circuit. This investigation will provide a better understanding of how the VQC structures behave under different initialization strategies for binary classification problems.

Variational quantum algorithms (VQAs), which classically optimize a parametrized quantum circuit to solve a computational task, promise to advance our understanding of quantum many-body systems and improve machine learning algorithms using near-term quantum computers. Prominent challenges associated with this family of quantum-classical hybrid algorithms are the control of quantum entanglement and quantum gradients linked to their classical optimization. Known as the barren plateau phenomenon, these quantum gradients may rapidly vanish in the presence of volume-law entanglement growth, which poses a serious obstacle to the practical utility of VQAs. Inspired by recent studies of measurement-induced entanglement transition in random circuits, we investigate the entanglement transition in variational quantum circuits endowed with intermediate projective measurements. Considering the Hamiltonian Variational Ansatz (HVA) for the XXZ model and the Hardware Efficient Ansatz (HEA), we observe a measurement-induced entanglement transition from volume-law to area-law with increasing measurement rate. Moreover, we provide evidence that the transition belongs to the same universality class of random unitary circuits. Importantly, the transition coincides with a “landscape transition” from severe to mild/no barren plateaus in the classical optimization. Our work may provide an avenue for improving the trainability of quantum circuits by incorporating intermediate measurement protocols in currently available quantum hardware.

Portfolio optimization under conditions of risk and market uncertainty remains a foundational challenge in financial economics and investment management. Classical optimization frameworks, including mean–variance models and stochastic programming approaches, often struggle with high-dimensional asset universes, nonlinear correlations, and dynamic market regimes. This paper examines quantum computing approaches to portfolio optimization under risk and market uncertainty, focusing on hybrid quantum–classical algorithms designed to enhance combinatorial optimization and probabilistic modeling. By integrating quantum annealing, variational quantum circuits, and quantum-enhanced sampling techniques with classical financial analytics, the study evaluates their potential to improve computational efficiency and scenario sensitivity. The analysis considers both theoretical advantages and practical limitations within the Noisy Intermediate-Scale Quantum (NISQ) era. The findings suggest that quantum-assisted optimization may complement traditional portfolio models by expanding feasible solution spaces and improving robustness under uncertainty, thereby contributing to more adaptive and resilient investment strategies.

In this work, we introduce the Quantum Adaptive Excitation Network (QAE-Net), a hybrid quantum-classical framework designed to enhance channel attention mechanisms in Convolutional Neural Networks (CNNs). QAE-Net replaces the classical excitation block of Squeeze-and-Excitation modules with a shallow Variational Quantum Circuit (VQC), leveraging quantum superposition and entanglement to capture higher-order inter-channel dependencies that are challenging to model with purely classical approaches. We evaluate QAE-Net on benchmark image classification tasks, including MNIST, FashionMNIST, and CIFAR-10, and observe consistent performance improvements across all datasets, with particularly notable gains on tasks involving three-channel inputs. Furthermore, experimental results demonstrate that increasing the number of variational layers in the quantum circuit leads to progressively higher classification accuracy, underscoring the expressivity benefits of deeper quantum models. These findings highlight the potential of integrating VQCs into CNN architectures to improve representational capacity while maintaining compatibility with nearterm quantum devices. The proposed approach is tailored for the Noisy Intermediate-Scale Quantum (NISQ) era, offering a scalable and feasible pathway for deploying quantum-enhanced attention mechanisms in practical deep learning workflows.

Quantum noise is the key challenge in Noisy Intermediate-Scale Quantum (NISQ) computers. Previous work for mitigating noise has primarily focused on gate-level or pulse-level noise-adaptive compilation. However, limited research efforts have explored a higher level of optimization by making the quantum circuits themselves resilient to noise. We propose QuantumNAS, a comprehensive framework for noise-adaptive co-search of the variational circuit and qubit mapping. Variational quantum circuits are a promising approach for constructing QML and quantum simulation. However, finding the best variational circuit and its optimal parameters is challenging due to the large design space and parameter training cost. We propose to decouple the circuit search and parameter training by introducing a novel SuperCircuit. The SuperCircuit is constructed with multiple layers of pre-defined parameterized gates and trained by iteratively sampling and updating the parameter subsets (SubCircuits) of it. It provides an accurate estimation of SubCircuits performance trained from scratch. Then we perform an evolutionary co-search of SubCircuit and its qubit mapping. The SubCircuit performance is estimated with parameters inherited from SuperCircuit and simulated with real device noise models. Finally, we perform iterative gate pruning and finetuning to remove redundant gates. Extensively evaluated with 12 QML and VQE benchmarks on 14 quantum computers, QuantumNAS significantly outperforms baselines. For QML, QuantumNAS is the first to demonstrate over 95% 2-class, 85% 4-class, and 32% 10-class classification accuracy on real QC. It also achieves the lowest eigenvalue for VQE tasks on H2, H2O, LiH, CH4, BeH2 compared with UCCSD. We also open-source TorchQuantum (https://github.com/mit-han-lab/torchquantum) for fast training of parameterized quantum circuits to facilitate future research.

We have developed two quantum classifier models for the ttH classification problem, both of which fall into the category of hybrid quantumclassical algorithms for Noisy Intermediate Scale Quantum devices (NISQ). Our results, along with other studies, serve as a proof of concept that Quantum Machine Learning (QML) methods can have similar or better performance, in specific cases of low number of training samples, with respect to conventional ML methods even with a limited number of qubits available in current hardware. To utilise algorithms with a low number of qubits — to accommodate for limitations in both simulation hardware and real quantum hardware — we investigated different feature reduction methods. Their impact on the performance of both the classical and quantum models was assessed. We addressed different implementations of two QML models, representative of the two main approaches to supervised quantum machine learning today: a Quantum Support Vector Machine (QSVM), a kernel-based method, and a Variational Quantum Circuit (VQC), a variational approach.

Quantum machine learning stands out as one of the most promising applications of quantum computing, widely believed to possess potential quantum advantages. In the era of noisy intermediate-scale quantum, the scale and quality of quantum computers are limited, and quantum algorithms based on fault-tolerant quantum computing paradigms cannot be experimentally verified in the short term. The variational quantum algorithm design paradigm can better adapt to the practical characteristics of noisy quantum hardware and is currently one of the most promising solutions. However, variational quantum algorithms, due to their highly entangled nature, encounter the phenomenon known as the “barren plateau” during the optimization and training processes, making effective optimization challenging. This paper addresses this challenging issue by researching a variational quantum neural network optimization method based on collective intelligence algorithms. The aim is to overcome optimization difficulties encountered by traditional methods such as gradient descent. We study two typical applications of using quantum neural networks: random 2D Hamiltonian ground state solving and quantum phase recognition. We find that the collective intelligence algorithm shows a better optimization compared to gradient descent. The solution accuracy of ground energy and phase classification is enhanced, and the optimization iterations are also reduced. We highlight that the collective intelligence algorithm has great potential in tackling the optimization of variational quantum algorithms.

Variational quantum algorithms (VQAs) are promising hybrid quantum-classical methods designed to leverage the computational advantages of quantum computing while mitigating the limitations of current noisy intermediate-scale quantum (NISQ) hardware. Although VQAs have been demonstrated as proofs of concept, their practical utility in solving real-world problems—and whether quantum-inspired classical algorithms can match their performance—remains an open question. We present a novel application of the variational quantum linear solver (VQLS) and its classical neural quantum states-based counterpart, the variational neural linear solver (VNLS), as key components within a minimum map Newton solver for a complementarity-based rigid-body contact model. We demonstrate using the VNLS that our solver accurately simulates the dynamics of rigid spherical bodies during collision events. These results suggest that quantum and quantum-inspired linear algebra algorithms can serve as viable alternatives to standard linear algebra solvers for modelling certain physical systems.This article is part of the theme issue ‘Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms’.

The noisy intermediate-scale quantum devices enable the implementation of the variational quantum circuit (VQC) for quantum neural networks (QNN). Although the VQC-based QNN has succeeded in many machine learning tasks, the representation and generalization powers of VQC still require further investigation, particularly when the dimensionality of classical inputs is concerned. In this work, we first put forth an end-to-end QNN, TTN-VQC, which consists of a quantum tensor network based on a tensor-train network (TTN) for dimensionality reduction and a VQC for functional regression. Then, we aim at the error performance analysis for the TTN-VQC in terms of representation and generalization powers. We also characterize the optimization properties of TTN-VQC by leveraging the Polyak-Lojasiewicz condition. Moreover, we conduct the experiments of functional regression on a handwritten digit classification dataset to justify our theoretical analysis.

Variational quantum eigensolver (VQE), aiming at determining the ground state energy of a quantum system described by a Hamiltonian on noisy intermediate scale quantum (NISQ) devices. However, the accuracy and trainability of the current VQE algorithm are significantly influenced due to the barren plateau (BP), the non‐negligible gate error and limited coherence time in NISQ devices. To tackle these issues, a gradient‐cost multiobjective alternate framework with variable ansatz is proposed. A theoretical framework is first proposed for VQE with variable ansatz (VA‐VQE) via alternately solving a multiobjective optimization problem defined by cost function values and gradient magnitudes and the original VQE problem. Then, a novel implementation method based on the double ε‐greedy strategy with the candidate tree and modified multiobjective genetic algorithm is proposed. As a result, the local optima are avoided both in ansatz and parameter perspectives, the BP phenomenon is alleviated, and the stability of output ansatz is enhanced. The experimental results indicate that this framework shows considerably average improvement of the error and stability by 59.6% and 78.8% compared with the state‐of‐the‐art cost‐criterion‐based VA‐VQE implementation, and 54.1% and 73.5% compared with the gradient‐criterion‐based counterpart, respectively, with lower quantum costs.

Advancements in noisy intermediate-scale quantum (NISQ) computing are steadily pushing these systems toward outperforming classical supercomputers on specific well-defined computational tasks. In this work we explore and control quantum chaos in NISQ systems using discrete-time quantum walks (DTQWs) on cyclic graphs. To efficiently implement quantum walks on NISQ hardware, we employ the quantum Fourier transform to diagonalize the conditional shift operator, optimizing circuit depth and fidelity. We experimentally realize the transition from quantum chaos to order via DTQW dynamics on both odd and even cyclic graphs, specifically 3- and 4-cycle graphs, using the counterintuitive Parrondo paradox strategy across three different NISQ devices. While the 4-cycle graphs exhibit high-fidelity quantum evolution, the 3-cycle implementation shows significant fidelity improvement when augmented with dynamical decoupling pulses. Our results demonstrate a practical approach to probing and harnessing controlled chaotic dynamics on real quantum hardware, laying the groundwork for future quantum algorithms and cryptographic protocols based on quantum walks.

The paradigm of variational quantum classifiers (VQCs) encodes classical information as quantum states, followed by quantum processing and then measurements to generate classical predictions. VQCs are promising candidates for efficient utilizations of noisy intermediate scale quantum (NISQ) devices: classifiers involving M-dimensional datasets can be implemented with only lceil log _2 M rceil qubits by using an amplitude encoding. A general framework for designing and training VQCs, however, is lacking. An encouraging specific embodiment of VQCs, quantum circuit learning (QCL), utilizes an ansatz: a circuit with a predetermined circuit geometry and parametrized gates expressing a time-evolution unitary operator; training involves learning the gate parameters through a gradient-descent algorithm where the gradients themselves can be efficiently estimated by the quantum circuit. The representational power of QCL, however, depends strongly on the choice of the ansatz, as it limits the range of possible unitary operators that a VQC can search over. Equally importantly, the landscape of the optimization problem may have challenging properties such as barren plateaus and the associated gradient-descent algorithm may not find good local minima. Thus, it is critically important to estimate (i) the price of ansatz; that is, the gap between the performance of QCL and the performance of ansatz-independent VQCs, and (ii) the price of using quantum circuits as classical classifiers: that is, the performance gap between VQCs and equivalent classical classifiers. This paper develops a computational framework to address both these open problems. First, it shows that VQCs, including QCL, fit inside the well-known kernel method. Next it introduces a framework for efficiently designing ansatz-independent VQCs, which we call the unitary kernel method (UKM). The UKM framework enables one to estimate the first known computationally-determined bounds on both the price of ansatz and the price of any speedup advantages of VQCs: numerical results with datatsets of various dimensions, ranging from 4 to 256, show that the ansatz-induced gap can vary between 10 and 20%, while the VQC-induced gap (between VQC and kernel method) can vary between 10 and 16%. To further understand the role of ansatz in VQCs, we also propose a method of decomposing a given unitary operator into a quantum circuit, which we call the variational circuit realization (VCR): given any parameterized circuit block (as for example, used in QCL), it finds optimal parameters and the number of layers of the circuit block required to approximate any target unitary operator with a given precision.

Variational quantum algorithms are promising candidates for delivering practical quantum advantage on noisy intermediate-scale quantum (NISQ) hardware. However, optimizing the noisy cost functions associated with these algorithms is challenging for system sizes relevant to quantum advantage. In this work, we investigate the effect of noise on optimization by studying a variational quantum eigensolver (VQE) algorithm calculating the ground state of a spin chain model, and we observe an abrupt transition induced by noise to the optimal solutions. We will present numerical simulations, a demonstration using an IBM quantum processor unit (QPU), and a theoretical analysis indicating the origin of this transition. Our findings suggest that careful analysis is crucial to avoid misinterpreting the noise-induced features as genuine algorithm results.

Variational Quantum Algorithms (VQAs) may be a path to quantum advantage on Noisy Intermediate-Scale Quantum (NISQ) computers. A natural question is whether noise on NISQ devices places fundamental limitations on VQA performance. We rigorously prove a serious limitation for noisy VQAs, in that the noise causes the training landscape to have a barren plateau (i.e., vanishing gradient). Specifically, for the local Pauli noise considered, we prove that the gradient vanishes exponentially in the number of qubits n if the depth of the ansatz grows linearly with n. These noise-induced barren plateaus (NIBPs) are conceptually different from noise-free barren plateaus, which are linked to random parameter initialization. Our result is formulated for a generic ansatz that includes as special cases the Quantum Alternating Operator Ansatz and the Unitary Coupled Cluster Ansatz, among others. For the former, our numerical heuristics demonstrate the NIBP phenomenon for a realistic hardware noise model.