The Noisy Intermediate-Scale Quantum (NISQ) era, defined by qubit counts in the hundreds and gate error rates that preclude long coherence times, imposes severe constraints on purely quantum computation. The central problem is that achieving practical quantum advantage for real-world tasks must overcome these hardware limitations, yet fault-tolerant quantum computers with full error correction remain years away. To address this, we present a systematic survey of hybrid classical-quantum computing models, in which classical high-performance resources orchestrate the workflow, perform iterative optimization, and apply error mitigation to noisy quantum processing units. Our review encompasses variational quantum algorithms, such as the variational quantum eigensolver and quantum approximate optimization algorithm; classical optimizers tailored for noisy quantum landscapes; error mitigation techniques including zero-noise extrapolation and probabilistic error cancellation; and circuit knitting that partitions large circuits across small QPUs. We find that these hybrid architectures have already demonstrated domain-specific utility in quantum chemistry for molecular ground-state estimation, in logistics and finance for combinatorial optimization, and in machine learning for generative and recommendation models. Notably, they often surpass purely classical or purely quantum approaches on carefully chosen benchmarks. The evidence indicates that hybrid classical-quantum systems are not a temporary stopgap but a durable paradigm. Their role will persist into the early fault-tolerant era, where small, logical qubits will similarly require classical coordination. Consequently, the field urgently demands the establishment of standardized hybrid performance benchmarks, the maturation of robust quantum-classical middleware capable of efficient transpiration and execution management, and holistic co-design frameworks that seamlessly integrate hardware constraints, software stack layers, and application-specific requirements from the outset.
The Noisy Intermediate-Scale Quantum (NISQ) era, defined by qubit counts in the hundreds and gate error rates that preclude long coherence times, imposes severe constraints on purely quantum computation. The central problem is that achieving practical quantum advantage for real-world tasks must overcome these hardware limitations, yet fault-tolerant quantum computers with full error correction remain years away. To address this, we present a systematic survey of hybrid classical-quantum computing models, in which classical high-performance resources orchestrate the workflow, perform iterative optimization, and apply error mitigation to noisy quantum processing units. Our review encompasses variational quantum algorithms, such as the variational quantum eigensolver and quantum approximate optimization algorithm; classical optimizers tailored for noisy quantum landscapes; error mitigation techniques including zero-noise extrapolation and probabilistic error cancellation; and circuit knitting that partitions large circuits across small QPUs. We find that these hybrid architectures have already demonstrated domain-specific utility in quantum chemistry for molecular ground-state estimation, in logistics and finance for combinatorial optimization, and in machine learning for generative and recommendation models. Notably, they often surpass purely classical or purely quantum approaches on carefully chosen benchmarks. The evidence indicates that hybrid classical-quantum systems are not a temporary stopgap but a durable paradigm. Their role will persist into the early fault-tolerant era, where small, logical qubits will similarly require classical coordination. Consequently, the field urgently demands the establishment of standardized hybrid performance benchmarks, the maturation of robust quantum-classical middleware capable of efficient transpiration and execution management, and holistic co-design frameworks that seamlessly integrate hardware constraints, software stack layers, and application-specific requirements from the outset.
Abstract The arrival of noisy intermediate-scale quantum (NISQ) devices defines an era in which quantum utility is fundamentally constrained by finite coherence times, limited qubit counts, gate infidelities, restricted connectivity, and measurement noise. Within this regime, hybrid quantum-classical frameworks have emerged as the dominant operational paradigm, particularly for quantum machine learning (QML). This review identifies a fundamental trade-off between expressivity, trainability, and noise resilience that governs the performance and scalability of NISQ-era QML systems. To contextualize these challenges, we introduce the resource-constrained iterative routing framework, a hardware-aware conceptual architecture that integrates topology-aware classical compression and distributed quantum orchestration to maintain circuit depths within device-dependent limits while routing representations toward symmetry-constrained Hilbert subspaces to mitigate noise-induced barren plateaus. Furthermore, we shift the discussion of quantum advantage away from asymptotic fault-tolerant complexity toward the notion of operational NISQ advantage based on finite-resource scaling, trainability preservation, and sampling complexity. We evaluate the practical resource efficiency of variational quantum algorithms, quantum kernel methods, and quantum neural networks across superconducting, trapped-ion, neutral-atom, and photonic platforms. The review also examines reproducibility challenges, emphasizing benchmarking against optimized classical baselines, and discusses strategies for managing the sampling overhead of error mitigation alongside distributed execution and quantum circuit compilation. Ultimately, NISQ in Practice positions hybrid QML as a physically constrained computational paradigm while outlining a pathway from contemporary noisy workflows toward early fault-tolerant quantum architectures.
Quantum computing has emerged as a transformative paradigm capable of addressing computational challenges beyond the reach of classical systems.In the current era of Noisy Intermediate-Scale Quantum (NISQ) devices, practical quantum hardware is characterized by limited qubit counts, short coherence times, gate errors, and the absence of full error correction.Despite these constraints, significant progress has been made in developing quantum machine learning (QML) techniques tailored to operate on near-term quantum processors.This paper presents a comprehensive review of Quantum Machine Learning in the context of NISQ computing.We first outline the fundamental principles of quantum computing, including qubits, superposition, entanglement, and quantum parallelism, followed by a detailed discussion of the NISQ computing model and its hardware limitations.The review then examines hybrid quantum-classical learning frameworks, variational quantum algorithms, quantum data encoding strategies, and noise mitigation techniques that enable practical implementation on existing platforms.Emerging applications across healthcare, finance, cybersecurity, climate modeling, and materials science are also surveyed to highlight the real-world potential of QML.Furthermore, key challenges such as scalability, trainability issues, and algorithmic robustness are analyzed, along with current research efforts aimed at overcoming these barriers.The paper concludes by identifying promising future directions for advancing QML toward quantum advantage.This review aims to provide researchers and practitioners with a structured understanding of the capabilities, limitations, and opportunities of quantum machine learning in the NISQ era .
Quantum computing has the potential to solve complex problems that are outside the scope of classical systems, making it a standard shifter. Current devices, which are considered as Noisy Intermediate-Scale Quantum (NISQ) systems, are insufficient in their practical utility due to hardware limitations, error rates, and qubit counts. The dynamic growths wanted to exceed the NISQ era and grow scalable quantum algorithms and architectures that can achieve useful quantum benefit in real-world applications are reviewed in this research. This research gives a comprehensive examination of the complications in scaling quantum systems, adjacent resource efficacy, qubit connectivity, and error correction. In addition to connecting architectural innovations like modular quantum computing, error-resilient circuit design, and distributed quantum systems, the paper provides new hybrid quantum-classical algorithms planned to exploit performance in noisy environments. Results from benchmarking in arenas like machine learning, optimization, and quantum chemistry show how these methods can provide short-term quantum benefit. Lastly, this paper proposes a research roadmap for quantum computing that highlights the requirement of co-designing algorithms, software, and hardware. The goal of this effort is to straighten the progress of quantum systems that can resolve problems in the real world at scale by addressing the inadequacies of NISQ devices and emphasizing scalable solutions.
Variational quantum algorithms (VQAs) have established themselves as a central computational paradigm in the Noisy Intermediate-Scale Quantum (NISQ) era. By coupling parameterized quantum circuits (PQCs) with classical optimization, they operate effectively under strict hardware limitations. However, as quantum architectures transition toward early fault-tolerant (EFT) and ultimate fault-tolerant (FT) regimes, the foundational principles and long-term viability of VQAs require systematic reassessment. This review offers an insightful analysis of VQAs and their progression toward the fault-tolerant regime. We deconstruct the core algorithmic framework by examining ansatz design and classical optimization strategies, including cost function formulation, gradient computation, and optimizer selection. Concurrently, we evaluate critical training bottlenecks, notably barren plateaus (BPs), alongside established mitigation strategies. The discussion then explores the EFT phase, detailing how the integration of quantum error mitigation and partial error correction can sustain algorithmic performance. Addressing the FT phase, we analyze the inherent challenges confronting current hybrid VQA models. Furthermore, we synthesize recent VQA applications across diverse domains, including many-body physics, quantum chemistry, machine learning, and mathematical optimization. Ultimately, this review outlines a theoretical roadmap for adapting quantum algorithms to future hardware generations, elucidating how variational principles can be systematically refined to maintain their relevance and efficiency within an error-corrected computational environment.
Abstract The emergence of noisy intermediate-scale quantum (NISQ) devices has fueled the development of hybrid quantum-classical algorithms for machine learning (ML), promising potential advantages over classical methods. However, the practical efficacy of these algorithms remains uncertain due to challenges like noise susceptibility, trainability issues, and a lack of rigorous benchmarking. This study presents a comprehensive empirical evaluation of two flagship hybrid algorithms the Variational Quantum Eigensolver (VQE) for classification and the Quantum Approximate Optimization Algorithm (QAOA) for combinatorial optimization against state-of-the-art classical baselines under equivalent resource constraints. Through extensive simulations incorporating realistic noise models, we demonstrate that while these quantum approaches achieve competitive performance (e.g., 91.2% accuracy for VQE on a binary classification task), they consistently underperformed optimized classical models like Support Vector Machines (94.8% accuracy) and require orders of magnitude more computational time. Furthermore, we provide empirical validation of critical barriers to scalability, including the barren plateau phenomenon and significant performance degradation under noise, even with advanced error mitigation. Our results indicate that for the tasks studied, current NISQ era algorithms do not yet provide a tangible quantum advantage. This work contributes a rigorous benchmarking framework and a critical, realistic assessment of the field, suggesting that future progress hinges on fundamental algorithmic innovations and hardware improvements rather than incremental refinements of existing paradigms.
Variational Quantum Algorithms are a major method of harnessing Noisy Intermediate-Scale Quantum devices to solve classically intractable problems in physics, chemistry, optimization, and machine learning. The current review offers a critical overview of VQAs, with special emphasis on their theoretical backgrounds, algorithmic structures, and application performance within realistic models of noise. Several key variants, such as the Variational Quantum Eigensolver, Quantum Approximate Optimization Algorithm, and Quantum Neural Networks, are compared in terms of noise robustness, ansatz archi- tecture, and hybrid quantum-classical methods of optimization. Benchmarks in fields such as quantum chemistry, finance modeling, and particles and nuclei physics reveal the poten- tial of VQAs in achieving practical quantum advantage. Fundamentally limiting factors 1such as barren plateaus, optimization overhead, and hardware constraints are considered in addition to mitigation strategies, including adaptive construction of ansatz, informed training in the presence of noise, and quantum error correction using QVECTOR. The paper is concluded with an overview of the scalability and future of VQAs in the direction of transitioning toward fault-tolerant quantum computing.