Quantum Machine Learning: A Hands-on Tutorial for Machine Learning Practitioners and Researchers

Dacheng Tao; Kaining Zhang; Min-Hsiu Hsieh; Naixu Guo; Patrick Rebentrost; Xinbiao Wang; Yang Qian; Yuxuan Du; Zhan Yu

REVIEW 4 major objections 4 minor 11 cited by

Quantum Machine Learning: A Hands-on Tutorial for Machine Learning Practitioners and Researchers

T0 review · 4 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This tutorial claims that a reader with a classical machine-learning background can learn to implement quantum kernels, quantum neural networks, and quantum transformers—and to judge when they could beat classical models.

desk verdict A useful, well-structured QML tutorial that needs a fix-it pass before it can deliver the hands-on promise. read the letter →

arxiv 2502.01146 v1 pith:URPZUQX2 submitted 2025-02-03 quant-ph cs.AIcs.LG

Yuxuan Du , Xinbiao Wang , Naixu Guo , Zhan Yu , Yang Qian , Kaining Zhang , Min-Hsiu Hsieh , Patrick Rebentrost

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Dacheng Tao

This is my paper · ORCID

classification quant-phcs.AIcs.LG

keywords quantummachinelearningkernelmethodsneuralnetworkstransformersblockencodingsingularvaluetransformationtrainabilitygeneralization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This tutorial argues that a reader coming from classical machine learning can, without prior quantum training, understand and implement the core models of quantum machine learning. The authors build this bridge by treating three model families—quantum kernel methods, quantum neural networks, and quantum transformers—each time starting from the classical version, giving the quantum construction, analyzing expressivity, trainability, and generalization, and ending with runnable code. The central premise is that quantum circuits can serve both as feature maps for kernels and as trainable models for QNNs, and that these quantum implementations may carry computational advantages over classical ones on specified tasks. If the tutorial works as intended, it supplies the AI community with a practical on-ramp to quantum computing while classical model scaling is hitting cost and physical limits.

What carries the argument

The workhorse is a data-dependent quantum circuit $U(x)$: it either acts as a feature map whose overlap $|\langle\phi(x)|\phi(x')\rangle|^2$ defines the quantum kernel, or as a parameterized ansatz whose measurement outcomes define a quantum neural network. For fault-tolerant algorithms, the tutorial relies on block encoding—placing a non-unitary matrix $A$ in the top-left corner of a larger unitary so that quantum singular value transformation (QSVT) can apply polynomial functions to singular values—as the unifying linear-algebra primitive. A named identity that does much of the theoretical work is the Fourier representation of angle-encoded quantum kernels, $k_Q(x,x') = \sum_{s,t\in\Omega} c_{st} e^{isx}e^{itx'}$, which shows how classical kernel structure emerges from the eigenvalue spectrum of the encoding gates.

What would settle it

Run every code listing in Chapters 2 through 5 exactly as printed; a NameError on `output_matrix` in Section 2.5.2 already shows one example fails, and if further listings fail the tutorial does not deliver hands-on learning. At the science level, if no dataset satisfies the paper's own criterion for potential advantage ($s_Q \ll n$ while $s_C \propto n$ and $g_{CQ} \sim \sqrt{n}$), the claimed scope for quantum prediction advantage would lack an instance.

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Extended reading notes

Core claim

The paper's central claim is that quantum machine learning—learning algorithms executed on quantum computers with potential advantages over classical implementations—can be presented from a classical ML foundation with theory and code together. Concretely, it claims that quantum kernels arise by encoding data into quantum states and evaluating their overlaps, that quantum neural networks are parameterized quantum circuits trained by classical optimizers, and that fault-tolerant quantum transformers can be built from quantum linear algebra, notably block encoding and quantum singular value transformation. It further asserts that the learnability of these models can be assessed along three axes—expressivity, trainability, and generalization—and that for quantum kernels a geometric criterion can indicate when a quantum kernel might beat all efficiently evaluable classical kernels. The tutorial's own framing is that it bridges 'it from bit' to 'it from qubit'.

Load-bearing premise

The tutorial's hands-on promise rests on the companion notebooks and printed code being complete and runnable; as published, the block-encoding example in Section 2.5.2 prints an undefined variable, so a reader following the text verbatim cannot execute it.

Editorial extensions

If this is right

A practitioner who finishes the tutorial should be able to implement a quantum kernel classifier on a reduced MNIST task and compare it to a classical SVM baseline.
The tutorial's theoretical sections turn the vague idea of quantum advantage into checkable criteria: compute $s_K(y) = y^{\top}K^{-1}y$ and the geometric difference $g_{CQ}$ to test whether a quantum kernel could beat classical kernels.
In the fault-tolerant setting, block encoding plus QSVT gives a concrete blueprint for quantum transformers, with the tutorial reporting quadratic speedups in runtime analysis.
Read-in and read-out bottlenecks are identified as the main caveat: exponential runtime speedups can be erased by the cost of loading classical data or extracting outputs, so advantage claims must count those costs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An implied next step, not pursued in the paper, is to apply the same classical-foundation to quantum-construction to theory to code template to other model families, such as diffusion models, graph neural networks, or recommender systems.
A reader could test the tutorial's completeness by executing every printed code block against a pinned software environment; the undefined variable in Section 2.5.2 suggests the companion notebooks, not the printed snippets, are the reliable deliverable.
The geometric-difference test could be run on larger real-world datasets to see whether the predicted separation between quantum and classical kernels actually materializes, rather than only on adversarially constructed labels.
If hardware continues to scale per the quoted roadmaps, the tutorial's NISQ-to-FTQC progression suggests that quantum kernels trained today could be 'upgraded' to fault-tolerant variants using the same block-encoding toolbox.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First thing you should know: this is a tutorial, not a research preprint. It does not claim new theorems or experiments. Its value is educational, and on that axis it is mostly good: the structure is sensible (classical foundation first, then quantum kernel, QNN, transformer chapters), the selection of topics is current, and the code demos cover the right ground. If you are an ML researcher wanting a map of QML, you could do much worse.

What it does well: the exposition is generally careful, with proofs for central results and references for the rest. The chapter on quantum kernels properly explains the geometry test from Huang et al. and gives a concrete dataset construction. The quantum transformer chapter includes runtime analysis and numerical evidence, which is a nice touch. The self-citations are present but not load-bearing; the content is backed by the broader literature.

Where it is soft: the hands-on claim. Section 2.5.2 prints `output_matrix` but never defines it, so the block-encoding demo cannot run as printed. That is a direct hit on the stated goal. There are also two math slips: Eq. (2.42) gives the POVM probability with a square, but it should be `Tr(ρ E_i)` (or `<ψ|E_i|ψ>`), and Example 3.8 expands `RX(x)` as `cos(x)|0> - i sin(x)|1>` instead of using `x/2`. The final kernel in that example is correct, so it is an intermediate algebra error, not a wrong result, but in a tutorial that is exactly where it hurts. The `qml.device("default.qubit", range(3))` call may also break in current PennyLane.

These are all fixable. The core mathematical and conceptual content is sound, and I did not find a load-bearing flaw beyond the broken code. For a tutorial, though, the errors are proportionally serious because the entire pitch is 'you can implement this.'

Bottom line: this is for ML practitioners and students who want an overview of QML with code. It is a legitimate teaching manuscript, but it needs a careful revision pass, checked by actually running the notebooks. I would send it to review for a teaching or survey venue, but not to a research journal. If the authors fix the code and the two equations, it would be a useful resource.

My recommendation: engage with it as a revise-and-resubmit at an appropriate venue; the errors are real but localized.

Referee Report

4 major / 4 minor

Summary. This manuscript presents a comprehensive tutorial on quantum machine learning (QML) aimed at readers with a classical machine-learning background. It covers foundational quantum computing concepts, quantum kernel methods, quantum neural networks, and quantum transformers, mixing formal definitions, theorem statements with proof sketches, and PennyLane code demonstrations. The stated goal is to enable hands-on learning and to bridge the gap between classical ML and QML, with a companion website hosting Jupyter notebooks.

Significance. If the presentation is corrected, this tutorial could serve as a valuable entry point for the QML community, particularly because it covers recent topics such as quantum transformers, trainability, and generalization, and because it attempts to connect classical and quantum formulations chapter by chapter. The inclusion of code demonstrations and a companion website is a useful pedagogical complement. However, the manuscript's value depends critically on the correctness of its mathematical exposition and the runnability of its code; several errors in both categories currently compromise the tutorial's central promise of hands-on, self-consistent learning.

major comments (4)

[§2.2.2, Eq. (2.42)] The POVM measurement probability is stated as Pr(i) = |<psi|E_i|psi>|^2. For a POVM element E_i, the Born rule gives Pr(i) = <psi|E_i|psi> (equivalently Tr(rho E_i) for a density matrix). The square is incorrect and will mislead readers about generalized measurements. This is a fundamental concept in the tutorial's measurement section and must be fixed.
[§3.2.2, Example 3.8, Eq. (3.25); §3.2.4, Example 3.11, Eq. (3.32)] The expansion of RX(x)|0> = exp(-i x sigma_x / 2)|0> is written as cos(x)|0> - i sin(x)|1>. The correct expansion is cos(x/2)|0> - i sin(x/2)|1>. The final kernel formula in Example 3.8, cos^2((x-x')/2), is consistent with the correct expansion, indicating a typographical slip, but this is exactly the kind of error that undermines a tutorial. The same half-angle omission appears in Example 3.11, Eq. (3.32), where the kernel is given as product_i cos^2(x_i - x'_i) rather than product_i cos^2((x_i - x'_i)/2).
[§2.5.2] The block-encoding code ends with print(np.real(np.round(output_matrix, 2))), but output_matrix is never defined or computed in the snippet. The example therefore cannot run as printed, directly violating the tutorial's hands-on promise. The authors should either define output_matrix, for instance by reconstructing the block-encoded matrix from the circuit state, or remove the print statement.
[§3.3.1, proof of Theorem 3.13] The proof relies on Algorithm 1 and Lemma 3.15, both of which assume input vectors with unit ell_1 norm (||r||_1 = 1). The Mercer feature map Phi_m obtained in the proof is normalized in the ell_2 norm only (assumed that ||Phi_m(x)|| = 1 for all x), and no argument is supplied for converting an ell_2-unit vector into an ell_1-unit vector while preserving the inner-product identity. As written, the universality proof is incomplete. The authors should clarify this step or provide a modified embedding that maps ell_2-unit vectors into ell_1-unit vectors.

minor comments (4)

[§2.5.1] The basis-encoding example calls qml.device("default.qubit", range(3)), passing wires as a positional argument; use wires=range(3) for consistency with the rest of the tutorial and with current PennyLane APIs.
[Throughout] There are several typographical errors, including seperable in Definition 2.2, Angel encoding in Table 3.2, and matrics in a remark in §2.3.2; a careful proofread is recommended.
[§2.3.1, Eq. (2.47)] The angle-encoding expression writes R_sigma(x_i)|0>^{⊗N} inside the tensor product; the state notation should be indexed per qubit, for instance |0>_i, to avoid confusion about the tensor product structure.
[§3.4, Eq. (3.68)] The derivation of the kernel estimation via U(x')U(x)^dagger includes an intermediate factor <0^{⊗N}|U(x')U(x)^dagger|0>^{⊗N}<0^{⊗N}|U(x')U(x)^dagger|0>^{⊗N}, which is a slightly abusive but understandable notation; consider clarifying the projector |0><0|^{⊗N} for readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper is an expository tutorial whose claims are supported by standard references, external learning theory, and self-contained derivations.

full rationale

The paper is a tutorial and makes no novel empirical predictions that could reduce to fitted inputs. Its central pedagogical claim—that a classical ML reader can learn QML—is not derived from its own equations. The mathematical content is standard: quantum kernels are defined and then checked for positive definiteness using Mercer's condition; Theorem 3.13 is proved from Mercer's theorem plus quantum-circuit universality rather than from its own conclusion. Generalization bounds are quoted or derived from the external learning-theory literature (Mohri; Huang et al. 2021a). The few self-citations (e.g., Guo et al. 2024a for state-preparation encoding and Hadamard-product block encodings, Qian et al. 2024 for adaptive shot allocation, Du et al. 2022a for quantum architecture search) are ancillary references to prior work, not the load-bearing justification of the tutorial's claims. Their stated lemmas are accompanied by proof sketches or references to independent results (Zhao et al. 2021; Gilyén et al. 2019), so they do not create a self-citation chain. The manuscript does contain technical errors (undefined 'output_matrix' in §2.5.2; RX expansion inconsistency in Example 3.8), but those are correctness defects, not circularity. No step in the paper's derivation chain is equivalent to its input by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The tutorial depends on standard mathematical and physical background, not on new postulates. No free parameters are fitted; no entities are invented.

assumptions (4)

domain assumption Postulates of quantum mechanics: state vectors have unit norm, evolution is unitary, measurement follows the Born rule.
Foundational for all quantum circuit definitions in Chapter 2; not proved in the tutorial.
standard math Mercer's condition: a continuous symmetric kernel is valid if it is positive definite.
Fact 3.1; used to assert quantum kernels are valid kernels via positive definiteness.
standard math Solovay-Kitaev theorem: any unitary can be approximated by a universal gate set with polylog overhead.
Fact 2.8; used to justify the circuit model's universality.
standard math Rademacher complexity and Talagrand contraction bounds for generalization.
Used in Chapter 3.3.2 to derive kernel generalization bounds; stated as Fact 3.16 from Mohri (2018).

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Cite this review

Pith. "Pith review of Quantum Machine Learning: A Hands-on Tutorial for Machine Learning Practitioners and Researchers." pith.science (2026). https://pith.science/paper/URPZUQX2

@misc{pith2026250201146,
  author       = {Pith},
  title        = {Pith review of: Quantum Machine Learning: A Hands-on Tutorial for Machine Learning Practitioners and Researchers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/URPZUQX2}},
  note         = {Machine review of arXiv:2502.01146}
}

read the original abstract

This tutorial intends to introduce readers with a background in AI to quantum machine learning (QML) -- a rapidly evolving field that seeks to leverage the power of quantum computers to reshape the landscape of machine learning. For self-consistency, this tutorial covers foundational principles, representative QML algorithms, their potential applications, and critical aspects such as trainability, generalization, and computational complexity. In addition, practical code demonstrations are provided in https://qml-tutorial.github.io/ to illustrate real-world implementations and facilitate hands-on learning. Together, these elements offer readers a comprehensive overview of the latest advancements in QML. By bridging the gap between classical machine learning and quantum computing, this tutorial serves as a valuable resource for those looking to engage with QML and explore the forefront of AI in the quantum era.

Figures

Figures reproduced from arXiv: 2502.01146 by the authors.

**Figure 1.1.** The paradigm between classical and quantum computing. The mechanisms between classical and quantum computing are very similar, where both of them involve input, computation, and output. In classical computing, the input refers to a bit-string, the computation part refers to the digital logic circuits, and the output also refers to a bit-string. In quantum computing, the input is a single- or multi-qubit state. The … view at source ↗

**Figure 1.2.** Different research directions in QML. QML can be categorized into four types based on the interplay of quantum (Q) and classical (C) systems: Q for Q (quantum algorithms for quantum data), Q for C (quantum algorithms for classical data), C for Q (classical algorithms for quantum data), and C for C (classical algorithms for classical data). This tutorial primarily focuses on the Q for C category. Beyond the role o… view at source ↗

**Figure 1.3.** Common quantum architectures and roadmaps from different quantum companies. known for their high coherence times, precise control over individual qubits, and full connectivity of all qubits (Bruzewicz et al., 2019). Moreover, Rydberg atom systems enable flexible qubit connectivity through highly controllable interactions (Morgado and Whitlock, 2021). Besides these architectures, integrated photonic quantum comput… view at source ↗

Figures from the paper (38 more)

**Figure 1.4.** Figure 1.4: Mechanisms of DNNs and QNNs. Both DNNs and QNNs follow an iterative approach. At each iteration, they take input data, process it through multiple layers, and produce an output prediction. The key difference between DNNs and QNNs is the way of implementing their lea…

**Figure 1.5.** Figure 1.5: The learnability of quantum machine learning models. equivalent of neural architecture search), advanced optimization techniques (Stokes et al., 2020), and pruning methods to reduce the complexity of quantum models (Sim et al., 2021; Wang et al., 2023a). Other areas…

**Figure 2.1.** Figure 2.1: The summarization of quantum gates. The table contains the abbreviation, the mathematical form, and the graph representation of a set of universal quantum gates. i represents the imaginary unit [PITH_FULL_IMAGE:figures/full_fig_p041_2_1.png]

**Figure 2.2.** Figure 2.2: The evolution of the single-qubit state decomposed into the quantum gates. The evolution from |a⟩ to |aˆ⟩ can be visualized using a quantum circuit diagram, as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p042_2_2.png]

**Figure 2.3.** Figure 2.3: The decomposition of the multi-qubit circuit U in the case of N = 3. Remark The CNOT gate plays a pivotal role in quantum computing due to its unique ability to generate entangled states, such as the Bell states and GHZ states presented in Examples 2.1&2.3. Besides, …

**Figure 2.4.** Figure 2.4: The evolution of quantum states based on Theorem 2.15. Quantum measurements In addition to quantum gates and quantum channels that manipulate quantum states, another special operation in quantum circuits is measurement [PITH_FULL_IMAGE:figures/full_fig_p049_2_4.png]

**Figure 2.5.** Figure 2.5: The quantum circuit diagram with measurement. The quantum measurements can be categorized into two types, i.e., projective measurements and positive operator-valued measures (Preskill, 1999; Nielsen and Chuang, 2011). The projective measurement, which is also called…

**Figure 2.6.** Figure 2.6: Example of basis encoding for the integer 6. Amplitude encoding Amplitude encoding is a technique that maps classical data into the amplitudes of a quantum state. Given a vector x = (x0, . . . , xi , . . . , x2N −1 ) ∈ C 2N containing complex values, we first apply …

**Figure 2.7.** Figure 2.7: Example of QRAM encoding for the dataset D = {2, 3}. 2.3.2 Quantum read-out protocols Quantum read-out refers to the process of translating the quantum state resulting from a quantum computation into classical data, enabling further processing, interpretation, or opt…

**Figure 2.8.** Figure 2.8: Quantum circuit for block encoding. method, which enables one to implement functions onto singular values of block-encoded matrices. 2.4.1 Block encoding For many computational problems, such as solving linear equations, we need to deal with a non-unitary matrix A. H…

**Figure 3.1.** Figure 3.1: Various distributions of data points. The left and middle panels show the cases where data points can and cannot be separated by a straight line. The right panel shows that the kernel function could map the linearly inseparable data points into the high dimensional l…

**Figure 3.2.** Figure 3.2: The paradigm of classical and quantum kernels. Both of the classical and quantum kernels embed the data points from data space X into high-dimensional space, and then compute the kernel as the inner product of feature maps. The quantum kernel leverages quantum circui…

**Figure 3.3.** Figure 3.3: Two methods for computing the inner product of the kernel. (a) Loschmidt echo test. (b) Swap test. The inner product between quantum states can be efficiently estimated on quantum computers using techniques such as Loschmidt echo test (Kusumoto et al., 2021) and SWAP…

**Figure 3.4.** Figure 3.4: The expressivity and generalization ability of quantum kernels. Expressivity concerns the size of the hypothesis space HQ represented by quantum kernels, where H refers to the whole hypothesis space. Generalization ability considers the learned hypothesis that could…

**Figure 3.5.** Figure 3.5: A flowchart for understanding the potential for quantum prediction advantage (Adapted from Huang et al. (2021a)). where ∥ · ∥∞ is the spectral norm of the resulting matrix and we assume Tr(KQ) = Tr(KC) = n. The geometric difference g(KC||KQ) can be computed on a cla…

**Figure 3.6.** Figure 3.6: T-SNE visualization of MNIST dataset of two classes ‘3’ and ‘6’. Steps 2&3: Define quantum feature mapping and building quantum kernel. We use angle embedding as the quantum feature mapping method. The quantum kernel is implemented as follows. 1 dev = qml . device (…

**Figure 3.7.** Figure 3.7: Visualization of quantum kernel matrix on 20 samples, equally drawn from two classes ‘3’ and ‘6’ in MNIST dataset. From the visualization, we observe a clear block structure: • Most of the elements in the top-left and bottom-right blocks, where samples belong to the …

**Figure 3.8.** Figure 3.8: The classification accuracy on test data as a function of the number of training samples. 3.5 Bibliographic Remarks The foundational concept of using quantum computers to evaluate kernel functions, namely the concept of quantum kernels, was first explored by Schuld e…

**Figure 4.1.** Figure 4.1: Illustration of the perceptron. Inputs are processed with a weighted linear combination followed by a non-linear activation to produce the output. spiring architectures ranging from convolutional neural networks (CNNs) (LeCun et al., 1989) and residual neural networ…

**Figure 4.2.** Figure 4.2: Illustration of a multilayer perceptron with two hidden layers. Dashed lines denote softmax operations. 4.1.2 Multilayer perceptron The multilayer perceptron (MLP) is a fully connected neural network architecture consisting of three components: the input layer, hidd…

**Figure 4.3.** Figure 4.3: Illustration of backpropagation when calculating the gradient of an MLP with one hidden layer. The index of sample a is omitted for simplicity. As illustrated in [PITH_FULL_IMAGE:figures/full_fig_p130_4_3.png]

**Figure 4.4.** Figure 4.4: Illustration of a QNN. The input state ρin is prepared using the operation Udata, followed by a variational quantum circuit (VQC) V (θ) and the measurement operation. 2024; Gao et al., 2024), researchers have devoted considerable effort to developing and implementin…

**Figure 4.5.** Figure 4.5: Illustration of a hardware-efficient circuit with two entanglement layers. model circuit, and the measurement. Input. The QNN uses quantum states ρin as input data. As shown in Table 4.2, QNNs can process both classical and quantum data. Specifically, the input sta…

**Figure 4.6.** Figure 4.6: Illustration of a quantum convolutional neural network [PITH_FULL_IMAGE:figures/full_fig_p141_4_6.png]

**Figure 4.7.** Figure 4.7: The quantum generator used in the quantum patch GAN, where each U(θℓ,n) ∈ U(2) is a trainable single-qubit unitary. To illustrate the training and sampling processes of QGAN, we present two examples based on the quantum patch and batch GANs proposed by Huang et al. (…

**Figure 4.8.** Figure 4.8: The main structure of the quantum batch GAN. The figure illustrates the process of generation and training using fake data. The oracle Uz for encoding latent vectors, the quantum generator UG(θ), and the quantum discriminator UD(γ) are applied sequentially to the ini…

**Figure 4.9.** Figure 4.9: Overview of the expressivity, generalization ability, and trainability of QNNs. The expressivity of the employed QNNs determines its hypothesis space H (solid blue ellipse). When H has a moderate size and encompasses the target concept (solid red star), QNNs can achi…

**Figure 4.10.** Figure 4.10: Influence of model complexity on generalization error. of complexity is crucial for achieving the best balance between training performance and generalization. Expressivity of QNNs In this chapter, we analyze the generalization error of QNNs through a specific mea…

**Figure 4.11.** Figure 4.11: The geometric intuition of covering number. Covering number concerns the minimum number of spherical balls with radius ϵ that occupy the whole space. Definition 4.10 (Covering number). The covering number N (U, ϵ, ∥ · ∥) denotes the least cardinality of any subset V…

**Figure 4.12.** Figure 4.12: T-SNE visualization of Wine dataset of the first two classes. Step 2: Data encoding. To encode the 13 attributes of the Wine dataset into a quantum system, we use angle encoding introduced in Section 2.3.1, followed by a layer of CNOT gates acting on neighboring qub…

**Figure 4.13.** Figure 4.13: The circuit diagram of the quantum classifier. Step 4: Training and evaluation of quantum classifier. With the data and circuit model ready, we now move to the optimization of the quantum classifier. The mean squared error (MSE) is used as the loss function. The goa…

**Figure 4.14.** Figure 4.14: The training curve of the quantum classifier. As demonstrated in [PITH_FULL_IMAGE:figures/full_fig_p173_4_14.png]

**Figure 4.15.** Figure 4.15: Samples in the dataset optdigits. Step 2: Building the classical discriminator. The discriminator is a classical neural network responsible for distinguishing real images from fake ones. It consists of fully connected layers with ReLU activations. The final output i…

**Figure 4.16.** Figure 4.16: The images generated by quantum patch GAN during training. 4.6 Bibliographic Remarks Quantum neural networks (QNNs) have emerged as a prominent paradigm in quantum machine learning, demonstrating potential in both discriminative and generative learning tasks. For d…

**Figure 5.1.** Figure 5.1: A standard Transformer architecture, showing an encoder on the left, and a decoder on the right. Image by Daniel Voigt Godoy under CC BY. The encoder processes the source sequence through multiple layers of multi-headed self-attention and feed-forward networks, augm…

**Figure 5.2.** Figure 5.2: Overview of the single-layer decoder-only quantum transformer. A quantum transformer consists of a self-attention and a feed-forward network sub-layer, incorporating residual connections with layer normalization. The inputs of the quantum transformer are block encod…

**Figure 5.3.** Figure 5.3: Scaling of the spectral norm ∥S∥ and the Frobenius norm ∥S∥F with ℓ for each model, displayed on logarithmic scales for both axes. For reference, the line y ∝ √ x is also shown. We use tokens in MMLU dataset and convert them to S. (Wq, Wk, Wv) for the large language …

**Figure 5.4.** Figure 5.4: Norms of weight matrices across open-source LLMs. The figure shows the maximum Frobenius norm and spectral norm values among weight matrices Wq, Wk, and Wv across the eight LLM models. 5.4 Code Demonstration We explain how self-attention works with a simple concrete …

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