Quantum Deep Learning: A Comprehensive Review

Yanjun Ji; Zhao-Yun Chen; Marco Roth; David A. Kreplin; Christian Schiffer; Martin King; Oliver Anton; M. Sahnawaz Alam

quant-phFeb 26, 2026

Quantum Deep Learning: A Comprehensive Review

Authors: Yanjun Ji, Zhao-Yun Chen, Marco Roth, David A. Kreplin, Christian Schiffer, Martin King, Oliver Anton, M. Sahnawaz Alam,

Organizations: Institute for Quantum Computing Analytics (PGI-12), Forschungszentrum Jülich, 52425 Jülich, Germany · Institute of Artificial Intelligence, Hefei Comprehensive National Science Center, Hefei, Anhui, 230088, P. R. China · Fraunhofer Institute for Manufacturing Engineering and Automation (IPA), Nobelstrasse 12, Stuttgart 70569, Germany · Heilbronn University of Applied Sciences, Max-Planck-Str. 39, 74081 Heilbronn, Germany · Institute of Neuroscience and Medicine (INM-1), Research Centre Jülich, Jülich, Germany · Helmholtz AI, Research Centre Jülich, Jülich, Germany · Munich Center for Mathematical Philosophy, Ludwig Maximilian University of Munich, Munich, Germany · Lichtenberg Group for History and Philosophy of Physics, University of Bonn, Bonn, Germany · Institut für Physik and Center for the Science of Materials Berlin (CSMB) Adlershof, Humboldt-Universität zu Berlin, Newtonstrß 15, 12489 Berlin, Germany · Ferdinand-Braun-Institut (FBH), Gustav-Kirchoff-Straße 4, 12489 Berlin, Germany · Institute for Automation of Complex Power Systems, RWTH Aachen University, Aachen, Germany · Jülich Supercomputing Centre, Forschungszentrum Jülich, 52425 Jülich, Germany · Faculty of Medical Engineering and Technomathematics, University of Applied Sciences Aachen, 52428 Jülich, Germany · Zentrum für Optische Quantentechnologien, Universität Hamburg, 22761 Hamburg, Germany · Institut für Quantenphysik, Universität Hamburg, 22761 Hamburg, Germany · The Hamburg Centre for Ultrafast Imaging, 22761 Hamburg, Germany · Theoretical Physics, Saarland University, 66123 Saarbrücken, Germany · Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui, 230026, P. R. China · CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China · Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, P. R. China · Origin Quantum Computing Technology (Hefei) Co., Ltd., Hefei, Anhui, 230026, P. R. China

Abstract

Quantum deep learning (QDL) explores the use of both quantum and quantum-inspired resources to determine when deep learning's core capabilities, such as expressivity, generalization, and scalability, can be enhanced based on specific resource constraints. Distinct from broader quantum machine learning, QDL emphasizes compositional depth at the pipeline level and the integration of quantum or quantum-inspired components within end-to-end workflows. This review provides an operational definition of QDL and introduces a taxonomy comprising four primary paradigms: hybrid quantum-classical models, quantum deep neural networks, quantum algorithms for deep learning primitives, and quantum-inspired classical algorithms. Theoretical principles are connected to advanced architectures, software toolchains, and experimental demonstrations across superconducting, trapped-ion, photonic, semiconductor spin, and neutral-atom systems, as well as quantum annealers. Claims of quantum advantage are critically assessed by distinguishing provable complexity-theoretic separations from empirical observations. The analysis characterizes trade-offs between model expressivity, trainability, and classical simulability, while systematically detailing the bottlenecks imposed by optimization landscapes, input-output access models, and hardware constraints. Applications are surveyed in domains encompassing image classification, natural language processing, scientific discovery, quantum data processing, and quantum optimal control, underscoring fair benchmarking against optimized classical counterparts and a comprehensive assessment of resource requirements. This review serves as a tutorial entry point for graduate students while guiding readers to specialized literature. It concludes with a verification-aware roadmap to transition QDL from near-term demonstrations to scalable and fault-tolerant implementations.

Figures & tables

Figure 1: Four paradigms of quantum deep learning (QDL), ordered left to right by increasing centrality of the quantum component to the end-to-end model. (a) Quantum-inspired algorithms: purely classical models in which the structure or training algorithms are motivated by quantum information theory. Quantum content is expressed through the model class, not through hardware-executed quantum states. (b) Hybrid quantum-classical models: a parameterized quantum circuit (PQC) with trainable gate parameters $\bm{\theta}$ is embedded as a differentiable module within a classical pipeline. The cycle denotes the classical outer optimization loop that updates $\bm{\theta}$ from measurement outcomes by minimizing a loss function $\mathcal{L}(\bm{\theta})$ . (c) Quantum algorithms for deep learning (DL) primitives: the quantum processing unit (QPU) acts as a specialized coprocessor to accelerate core subroutines, e.g., linear-system solvers such as HHL and the quantum singular value transformation (QSVT), within an overarching classical model under explicitly stated access and resource assumptions. (d) Quantum deep neural networks: deep, hierarchical quantum circuits designed for end-to-end quantum information processing, e.g., feature extraction directly from quantum data. $R_{x}(\theta_{i})$ and $R_{z}(\theta_{i})$ : single-qubit rotation gates; $X$ : Pauli- $X$ gate; $H$ : Hadamard gate.

Figure 2: The hybrid quantum-classical learning loop. Classical data $x$ are encoded into an input state $\rho_{\mathrm{in}}(x)$ via the data-encoding unitary $S(x)$ . A parameterized quantum circuit (PQC) $U(\bm{\theta})$ is executed, measurements of an observable yield finite-shot statistics used to estimate a loss function (e.g., expectation value $f(x;\bm{\theta})$ ), and a classical optimizer updates $\bm{\theta}$ by minimizing an objective built from these estimates. The loop repeats until convergence to an optimal parameter set $\bm{\theta}^{*}$ .

Figure 3: Taxonomy of hybrid quantum-classical architectures across four paradigms. (a) Placement and composition. A classical encoder compresses the input $x$ to a latent code $z$ , which is embedded via $S(z)$ ; this encoding of $z$ rather than the raw data $x$ directly alters the induced hypothesis class. An interleaved-exchange variant (lower) routes $x$ through alternating quantum modules $Q_{1},Q_{2},\ldots$ and classical layers where each $Q_{i}$ subsumes the full encode-process-measure pipeline $S(\cdot)\to U(\bm{\theta}_{i})\to\mathcal{M}$ of the upper subpanel. (b) Hierarchical and geometric structure. Quantum convolutional neural network (QCNN, upper): alternating trainable-unitary convolutional and measurement-based pooling layers act on $\rho_{\mathrm{in}}$ , followed by a fully connected final mixing layer; Graph models (lower): node and edge unitaries $U_{v}$ , $U_{e}$ implement permutation-equivariant message passing on $G{=}(V,E)$ . (c) Interfaces and encoding patterns. Quanvolution (upper): local PQCs process image patches $x_{i}$ and output a spatial feature map for classical aggregation. Data reuploading (lower left): interleaving $S(x)$ with layers $U(\bm{\theta}_{l})$ implements a Fourier-feature expansion. Independent and multi-channel (lower right): independent channels concatenate per-channel measurement outcomes classically, whereas in multi-channel, joint encoding applies coherent premeasurement mixing, enabling coherent cross-channel mixing in the quantum feature map prior to measurement. (d) Objective-driven paradigms. Training loop (upper): a classical backbone, quantum primitive $U(\bm{\theta})$ , and task objective jointly update $\bm{\theta}$ ; instantiations include quantum generative adversarial networks (QGAN) discriminators, Boltzmann models, and reinforcement learning (RL) value functions. Deployment loop (lower): $U(\bm{\theta}^{*})$ operates at trained and fixed parameters under a sampling bottleneck on quantum devices, producing scores, samples, or actions.

Paradigm	Quantum-module role	Inductive bias	Key refs.
Placement and composition (graph location)
PQC head	terminal readout / classifier	encoding-induced hypothesis restriction; restricted readout	[ 237 , 719 ]
Transfer hybrid	quantum head on frozen classical features	representation bottleneck; low-dimensional embedding	[ 546 , 162 ]
Interleaved exchange	multiple quantum blocks separated by classical processing	non-sequential composition; measurement-mediated feature exchange across modules	[ 500 , 525 , 299 , 398 ]
Hierarchical and geometric structure (symmetry + locality priors)
QCNN	hierarchical pooling	multiscale locality via pooling; parameter tying	[ 179 , 331 ]

Table 1: Representative taxonomy of hybrid quantum-classical models. Architectures are classified by the computational role of the quantum circuit and the primary structural constraint that shapes the hypothesis class and associated estimator family. Figure 3 depicts canonical core motifs for each paradigm; the table lists representative instantiations and closely related variants. Abbreviations: PQC, parameterized quantum circuit; QCNN, quantum convolutional neural network; QNN, quantum neural network; RL, reinforcement learning; QGAN, quantum generative adversarial network.

Figure 4: Fundamental tensor network (TN) architectures. Shown are matrix product state (MPS; also known as tensor train), projected entangled pair state (PEPS), tree tensor network (TTN), and the multiscale entanglement renormalization ansatz (MERA). The top row illustrates increasing tensor order (scalar, vector, matrix, tensor) as a visual legend for node types. Adapted from Berezutskii et al. [83] .

Figure 5: Resource complexities under an explicit access model

\mathcal{A}

. (a) three physical access regimes, including single-copy prepare-and-measure; collective access, where

k

copies of

|\psi\rangle

per query are processed jointly via quantum memory (

k

denotes the number of copies jointly processed per example instance); and coherent oracle or QRAM-style state access. (b)

\mathcal{A}

constrains hypothesis-class complexity

\mathcal{C_{H}}

, which in turn mediates trade-offs among three consumable resource axes: example complexity

\mathcal{C_{N}}

, measurement complexity

\mathcal{C_{S}}

(total circuit repetitions, i.e., shots

\times

settings, typically incurred per training example for a fixed estimator precision), and computational complexity

\mathcal{C_{T}}

. The number of total state preparations per objective evaluation is

\mathcal{C_{N}}\times\mathcal{C_{S}}

and scales as

k\penalty\ \mathcal{C_{N}}\times\mathcal{C_{S}}

under collective access.

Figure 6: Five-layer implementation stack for QDL, illustrating top-down requirements flow and bottom-up constraint propagation.

Modality	Primitive	Connectivity	QDL bottleneck
Superconducting	digital gates	sparse 2D (routing)	routing overhead, drift/crosstalk
Trapped ions	digital gates	all-to-all (in-modular); modular links	wall-clock throughput
Neutral atoms	digital gates + analog	reconfigurable between shots	entangling error, atom loss
Photonics	optical sampling; CV	interferometric meshes; delay networks	optical loss, calibration
Annealers	analog sampling	fixed graph (minor embedding)	embedding, non-equilibrium bias
Semiconductor spin	digital gates	local coupling	variability, coupling range

Table 2: Modality-level hardware characteristics most relevant to QDL feasibility and codesign. Connectivity refers to native two-qubit (or native entangling) interaction structure; QDL bottleneck denotes a typical dominant limitation for end-to-end hybrid training at current scales. CV: continuous-variable. For annealers, minor embedding refers to mapping a logical problem graph onto the fixed hardware graph.

Software	Classical Simulation			Hybrid Workflow			QPU Integration
Software	GPU-enabled	HPC-enabled	TN	PyTorch	JAX	Keras	Circuit-based	Photonics
PennyLane	✓	✓	✓	✓	✓	✓	✓	✓
TensorFlow Quantum	✓	✓	$\times$	$\times$	$\times$	✓	✓	$\times$
Qiskit Machine Learning	✓	✓	✓	✓	$\times$	$\times$	✓	$\times$
CUDA-Q	✓	✓	✓	✓	$\times$	$\times$	✓	$\times$
sQULearn	✓	$\circ$	$\times$	✓	$\times$	$\times$	✓	$\times$

Table 3: Comparison of ten mainstream QDL software frameworks. The evaluation spans classical simulation capabilities, integration with classical libraries for hybrid workflows, and support for diverse quantum hardware paradigms. GPU-enabled: officially supported GPU-accelerated simulation backend; HPC-enabled: documented multi-node and distributed execution; TN: documented tensor network (TN) approximate simulators integrated in the stack.

Figure 7: Taxonomy of QDL application regimes by data interface. Classical data pipelines embed classical features into quantum circuits and are typically bottlenecked by state-preparation overheads. Scientific discovery integrates quantum modules into label-expensive workflows with costly forward models for both classical and quantum data, where sample efficiency is paramount. Quantum data pipelines operate directly on quantum-native states or processes under an explicit access model, with bottlenecks set by the readout interface and the availability of coherent quantum memory. The lower panel summarizes the four-pillar evaluation protocol used in Sec. V.2 .

Pillar	Reporting axes	Primary failure modes
1. Empirical performance Does it improve quality under matched evaluation?	– Task-specific metric and calibration – Learning curves versus data and compute – Robustness across seeds and splits – Sample-efficiency diagnostics	– Under-tuned or outdated baselines – Unmatched hyperparameter optimization – Hidden pre or postprocessing – Selective reporting of best runs
2. Resource complexity Is it more efficient under end-to-end accounting?	– Total shots and QPU calls – Wall-clock breakdown and time-to-target – Energy and monetary cost boundary – Compiled depth and two-qubit counts – Classical coprocessing and communication	– Reporting circuit metrics without wall-clock cost – Incommensurate access or readout interfaces or training interfaces (e.g., classical backpropagation) – State-preparation and encoding omitted – Missing classical surrogate checks
3. Practical feasibility Can it be trained and executed reliably on quantum hardware?	– Training stability and failure rate – Verification method and its cost – Inference throughput and latency – Compilation overheads	– Optimizer stagnation and seed sensitivity – Shot-noise dominated updates – Unsupported intractability
4. Domain alignment Is the quantum component structurally matched to the task?	– Architecture class and symmetry constraints – Modality and interface realism – Structure-sensitive ablations and controls – Inductive-bias diagnostics	– Access-model mismatch to the deployment setting – Baselines that ignore the same structure – Claims of inductive bias without controls

Table 4: A four-pillar protocol for rigorous quantum-classical comparison in finite-size empirical studies. Claims are evaluated under an explicit access and readout interface and an end-to-end resource contract

\mathcal{R}

(Definition 3). Mechanisms underlying limitations are treated in Sec. III.7 .

Figure 8: Strategic roadmap for QDL, illustrating the coupled evolution of four dimensions: hardware scale, error management, algorithmic sophistication, and verification rigor. We distinguish three capability regimes by heuristic operation budgets and error-handling assumptions [ 228 , 655 ] . Here, an “elementary operation” means one primitive physical action in a single compiled circuit execution (one QPU call), counting each native single-qubit gate, two-qubit gate, and single-qubit measurement once (summed over all qubits). Under this convention: (i) the near-term NISQ regime spans

\sim 10^{4}

–

10^{6}

elementary operations per circuit execution, dominated by hybrid training and error mitigation; (ii) an early fault-tolerant regime spans

\sim 10^{6}

–

10^{9}

, enabling deeper circuits and explicitly QEC-aware training protocols; and (iii) an application-scale fault-tolerant regime spans

\sim 10^{9}

–

10^{12}

, where fully fault-tolerant quantum deep architectures and resource-intensive algorithmic subroutines become feasible.

Quantum Machine Learning

Quantum machine learning uses quantum circuits and quantum data to build learning algorithms and study their computational and statistical limits.

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Jun 30, 2026J/K move · Enter open · S save

Beyond the Expressivity-Trainability Paradox: A Dynamical Lie Algebra Perspective on Navigating Barren Plateaus in Quantum Machine Learning

Kung-Ming Lan, Edward Huang

Department of Industrial Engineering and Enterprise Information, Tunghai University, Taichung 40704, Taiwan · Department of Industrial and Systems Engineering, 3306 Shelby Center, Auburn University, Auburn, AL, USA

Quantum Machine Learning Barren Plateaus Variational Quantum Circuits Quantum Neural Networks

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As Quantum Machine Learning (QML) transitions toward practical implementation, the field faces a critical architectural bottleneck that challenges the fundamental assumptions of classical statistical learning theory. In classical deep learning, increasing model capacity typically risks overfitting. However, this study advances a counter-intuitive paradigm: unstructured contemporary QML architectures suffer from a profound state of quantum underfitting, driven by the "expressivity-trainability paradox." We demonstrate that the vast Hilbert space capacity of Parameterized Quantum Circuits (PQCs)-traditionally chased as the source of quantum advantage is the direct mathematical cause of Barren Plateaus (BPs), where gradient landscapes become exponentially flat. By synthesizing recent breakthroughs in Dynamical Lie Algebras (DLAs) and Geometric QML, we establish a comprehensive framework linking the algebraic dimension of circuit generators to their optimization dynamics. Furthermore, we empirically validate this framework on a non-linear binary classification task, illuminating a uniquely quantum manifestation of the bias-variance tradeoff: while unstructured architectures achieve near-perfect training accuracy via unscalable parameterization (quantum overfitting), embedding group-theoretic geometric priors acts as a structural regularizer. By restricting the DLA growth to a polynomial regime, our symmetry-preserving approach sacrifices raw memorization capacity to guarantee scalable, gradient-rich training landscapes, offering a robust roadmap for "Trainability-by-Design" in scalable quantum neural networks.

Abstract

Quantum Machine Learning

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