Deep Learning in Classical and Quantum Physics

Marcin P{\l}odzie\'n; Timothy Heightman

REVIEW 5 major objections 7 minor 1 cited by

Deep Learning in Classical and Quantum Physics

T0 review · 5 major / 7 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read These lecture notes aim to give graduate students a complete, detailed introduction to deep learning for quantum applications.

desk verdict Readable, honest lecture notes, but several concrete equation errors (RBM partition function, KL decomposition) undermine the promised 'complete, detailed' pedagogy; fixable, but not yet ready for students. read the letter →

arxiv 2508.10666 v1 pith:FEZABY5P submitted 2025-08-14 quant-ph cs.AIcs.NEphysics.comp-ph

Timothy Heightman , Marcin P{l}odzień This is my paper

classification quant-phcs.AIcs.NEphysics.comp-ph

keywords deeplearningquantumphysicsneuralnetworksunsupervisedgenerativemodelsstatetomographyvariationaleigensolverIsingmodel

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

These lecture notes set out to be a complete, detailed introduction to deep learning for graduate students in physics, mathematics, engineering, and computer science, with quantum science as the target application area. The substantive claim is that deep learning can be taught as a transferable sequence of skills — building neural networks as tunable nonlinear functions, training them with gradient descent and automatic differentiation, and applying them to high-dimensional physical problems such as phase classification, generative modelling, and quantum state reconstruction. The notes also argue that deep learning can locate structure in data (for example, estimating the critical temperature of the 2D Ising model without using an order parameter) but does not by itself provide physical understanding, a boundary illustrated with the Chinese Room experiment. A sympathetic reader would come away with a concrete sense of when and how to use deep learning in quantum problems and what pitfalls to avoid; the text itself invites readers to report typos in its formulae.

What carries the argument

The load-bearing machinery is the layered neural network $f_\theta(x) = h^{(L)} \circ \cdots \circ h^{(1)}(x)$, where each layer applies a linear map (weight matrix and bias) followed by a nonlinear activation. Training is driven by gradient descent on a loss function, with gradients computed by automatic differentiation (backpropagation), and the reparametrisation trick extends differentiability to sampling from latent distributions, enabling VAEs, GANs, restricted Boltzmann machines, and normalising flows. These components carry the notes' argument because every later quantum application — phase classification, state representation, and tomography — is framed either as regression/classific

What would settle it

A reader can settle the central claim by opening the notes and trying to reproduce the examples. Eq. (3.24) defines the RBM partition function as $Z = \sum_z E(z)$; Boltzmann normalisation requires $Z = \sum_z e^{-E(z)}$, so the stated $p(z) = e^{-E(z)}/Z$ is not a valid probability distribution. The text after Eq. (3.8) says the KL divergence equals 1 when distributions are identical; the correct value is 0. Eq. (2.5) gives mismatched matrix dimensions and a bias index. Any one of these checks shows the notes are not yet complete and detailed as written.

Watch

Extended reading notes

Core claim

The notes aim to establish a pedagogical result: a graduate student can go from no familiarity with neural networks to working knowledge of deep learning for quantum problems by following a single progressive sequence. The chain of machinery is a neural network as a tunable nonlinear function $f_\theta$; the universal approximation theorem as the licence to use such functions; backpropagation and automatic differentiation as the training machinery; and the reparametrisation trick as the bridge to probabilistic generative models. On the quantum side, the claim is that these tools apply to concrete tasks: three-qubit entanglement classification, phase-diagram discovery with autoencoders, varia

Load-bearing premise

The promise that these notes form a complete, detailed, self-contained description rests on the assumption that the printed equations and numerical examples can be trusted and reproduced as written; as published, the RBM partition function in Eq. (3.24) ($Z = \sum_z E(z)$), the t-SNE KL-divergence statement after Eq. (3.8), and the layer dimensions in Eq. (2.5) contain errors, and the hands-on examples include no code or data.

Editorial extensions

If this is right

A student who works through the sequence can implement standard deep-learning models (multilayer perceptrons, CNNs, autoencoders, VAEs, GANs, RBMs, normalising flows) and deploy them on physics data.
The confidence-drop method gives a model-independent way to estimate a critical temperature when no order parameter is known, benchmarked on the 2D Ising model and extendable to other transitions.
Quantum applications such as variational quantum eigensolvers, QAOA, neural-network quantum states, Hamiltonian learning, and photonic tomography are cast as concrete deep-learning tasks, so the same training principles apply across them.
The notes' Chinese Room argument sets expectations: a deep-learning model can generalise and predict, but on its own it does not supply causal understanding of the underlying physics.
The notes identify overfitting, interpretability, and noise sensitivity as the main practical constraints a quantum scientist must mitigate when applying deep learning.

Reading between the lines

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

The notes leave implicit that the confidence-dip method for estimating $T_c$ is a general recipe: any classifier trained on well-separated phases could locate an unknown transition by monitoring where its softmax confidence collapses.
Because quantum mechanics is itself a probabilistic theory, the reparametrisation trick used for VAEs is a natural bridge to quantum applications; the notes do not pursue reparametrisations tailored to quantum state manifolds.
If the promised code and datasets were supplied, the same chapter sequence could be reused as a template for other data-intensive sciences, not only quantum physics.
A testable extension suggested by the notes' structure is to benchmark neural-network quantum states against normalising-flow tomography on the same small system to compare sample efficiency and reconstruction fidelity.

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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

Dear [Colleague],

Here's my take on arXiv:2508.10666. It's explicitly lecture notes, not a research paper, and that's the right frame. There is nothing new here—universal approximation, backprop, CNNs, VAEs, GANs, RBMs are all textbook—but the notes are readable, the order is sensible, and the physics connections (Ising, Onsager, later quantum chapters) are what will draw people in. The Chinese Room section is a nice non-technical anchor, and the reparametrization trick / Gibbs sampling boxes are genuinely pedagogically helpful.

The soft spots are real and load-bearing for a teaching document. Eq. (3.24) defines Z = Σ E(z) instead of Z = Σ exp(−E(z)). That's not a harmless typo: the RBM derivation that follows (Eqs. 3.36–3.39) factors the hidden sum as if Z = Σ exp(−E(v,h)), which is inconsistent with the printed definition. A student who implements Eq. (3.24) will compute nonsense gradients. Eq. (2.19) gives KL = H + Lce, but with H defined as Shannon entropy, the correct relation is KL = Lce − H. Box 2.8 repeats the same mistake. Eq. (2.5) has mismatched matrix dimensions and bias index. These are all individually correctable, but together they undermine the central claim that the notes are a 'complete, detailed description' you can work through.

I also could not find the quantum chapters (4–5) in the text I was given. The table of contents lists them, but the content isn't in this version. For quant-ph readers that's the entire point. If they're simply missing from the posted version, that's a serious incompleteness; if they're present but were cut off in my copy, I can't evaluate them. Either way, the version under review does not deliver the advertised scope.

The citation pattern is fine. They cite the original literature (Hinton, Kingma, Goodfellow, Onsager) and don't over-rely on self-citations.

My verdict: this deserves a serious referee, but the authors need to do a careful proofreading pass. The errors are concrete and will mislead the target reader. After corrections, I would happily point students toward it. As is, I'd hold it back.

Best.

Referee Report

5 major / 7 minor

Summary. This manuscript is a set of lecture notes, arXiv:2508.10666, aiming to provide a graduate-level, self-contained introduction to deep learning and its applications in classical and quantum physics. The first three chapters cover neural-network anatomy, training, classification examples (MNIST, CIFAR-10), the 2D Ising model, unsupervised methods (PCA, t-SNE, autoencoders, VAEs, GANs, restricted Boltzmann machines, normalising flows), and some physics-informed concepts. Later chapters summarize quantum mechanics and deep-learning techniques in quantum science. The paper makes no claim to new research results; its central claim is pedagogical: that a student working through the notes can acquire a correct, hands-on command of deep learning and judge when to apply it in quantum problems. The reviewed chapters contain several concrete mathematical errors, so the central claim is not met as printed.

Significance. If the equation-level errors were corrected, these notes would fill a useful niche: they connect deep-learning formalism to statistical physics and quantum applications at a level appropriate for graduate students. The organizational structure is clear, and the conceptual discussion—e.g., the Chinese-room thought experiment, the distinction between points and probability distributions in VAEs, and the motivation for energy-based models—is a genuine strength. The paper does not present original research, so circularity is not a concern; the concern is reliability of the reproduced material. No code, data, or machine-checked derivations are supplied, so the 'hands-on' promise depends entirely on the printed equations and figures. Because several load-bearing equations are wrong or internally inconsistent, the notes cannot currently be used as a trustworthy teaching resource.

major comments (5)

[§3.6, Eq. (3.24)] The partition function is defined as Z = Σ_z E(z), omitting the exponential Boltzmann weight. This contradicts Eq. (3.23), which correctly has p(z) = Z^{-1} exp(-E(z)). The subsequent derivation of P(v) in Eqs. (3.36)–(3.39) factors sums of exp(...) terms, which is impossible if Z were defined as in Eq. (3.24). The gradient formula (3.42) and Box 3.4 use the correct Z = Σ_{v,h} exp(-E(v,h)), so the text is internally inconsistent. A student implementing the equations literally will obtain nonsensical normalizations and gradients. This is a load-bearing error for the RBM section and for the 'complete, detailed description' promised in the preface.
[§2.2.1, Eq. (2.19) and Box 2.8] The KL divergence is given as KL(p||q) = H + L_ce, where H = -Σ p log p and L_ce = -Σ p log q. The correct identity is KL = Σ p log(p/q) = Σ p log p - Σ p log q = L_ce - H. The sign error is not purely typographical: Box 2.8 explains the decomposition in detail and repeats it, and the text uses the one-hot special case (where H=0) to obscure the general error. This undermines the explanation of why cross-entropy is the appropriate classification loss. The error appears twice, so it is unlikely to be a one-off typo.
[§2.1.1, Eqs. (2.5)–(2.7)] The layer transformation is dimensionally inconsistent. Eq. (2.5) states W^{(l)} is an N^{(l-1)} × N^{(l)} matrix and writes z ← W^{(l)}z + b^{(l-1)} with z ∈ R^{dim(l)}. If z has dimension N^{(l)}, then W^{(l)}z has dimension N^{(l-1)}, not N^{(l)}; if the layer output h^{(l)} should have N^{(l)} neurons, the weight matrix must have shape N^{(l)} × N^{(l-1)}. Eq. (2.6) repeats the same mixing of indices, while Eq. (2.7) uses sums over N_l and N_{l-1} that imply the correct convention is W^{(l)}: R^{N_{l-1}} → R^{N_l}. The §2.3 feed-forward shapes (e.g., X W^{(1)} with W^{(1)} of shape m × N^{(1)}) follow yet another convention. This makes it impossible for a student to verify the matrix multiplications by the stated definitions.
[§3.1.2, Eq. (3.5) and Eq. (3.8)] Two errors affect the t-SNE explanation. First, the conditional probability p(i|j) in Eq. (3.5) uses σ_i in the numerator but σ_j in the denominator; the standard formulation uses a single bandwidth for the reference point j. Second, the text after Eq. (3.8) says 'When these distributions are identical (L = 1), we get our closeness property perfectly!'—but the KL divergence is 0 when p = q, and L is minimized, not equal to 1. Both errors are conceptual, not just notational, and they directly concern a method presented as a core unsupervised tool.
[§2.3, §2.5.3] The numerical demonstrations are not reproducible from the text. No code, data, or random seeds are provided for the MNIST (Figs. 2.7–2.8), CIFAR-10 (Figs. 2.10, 2.13–2.15), or 2D Ising (Fig. 2.20) examples. Hyperparameters are incomplete or inconsistent: the MNIST experiment in §2.3.1 states η = 50, but §2.3.2 says 'same learning rate η = 1'; Fig. 2.7's caption says the input layer contains '282 nodes' where the text says 784. A student cannot reconstruct the claimed accuracies (56%, 95%, 82%, etc.) from the information given. This directly violates the preface's promise of 'hands-on examples' and weakens the pedagogical claim.

minor comments (7)

[§2.2.2, Eq. (2.25)] The gradient-descent update is written as θ_{t+1} ← θ_t − η L(θ_t)/∂θ_t. The denominator should be ∂L(θ_t)/∂θ_t. As printed, the expression is not a valid derivative.
[§2.3.1, Eq. (2.30)] The bias update line read 'b ← b − η ∂L/∂b = W − η ...'; the right-hand side should start with b, not W. This is a typo but confusing in a derivation.
[§2.5.3] The Ising-model deep-learning demonstration does not cite the original works on machine-learning identification of phases (e.g., Carrasquilla and Melko, or related studies). For a set of lecture notes, this is a notable citation gap.
[§3.6, Algorithm 4] The algorithm's input line says current RBM weights θ = {a, b, J}, while the text and Box 3.4 use W_{ij} for the coupling matrix. The notation should be unified.
[§3.1.2, Eq. (3.7)] The t-SNE kernel is written as q_{ij} = (1 + ||y_i - y_j||^2)^{-1} / Σ_{k≠l}(1 + ||y_k - y_l||^2)^{-1}. The denominator should be normalized over ordered pairs (k,l) with k≠l, and the text should note that the numerator is the unnormalized Student-t kernel before division. This is minor but worth a clarifying sentence.
[Various] There are many typographical and terminology issues: 'Principle Component Analysis' should be 'Principal Component Analysis'; 'Bolztmann' and 'Lecunn' are misspelled; Box 3.5 is labelled 'Box 3.4' in the text; 'mathematica' appears uncapitalized. None of these block the main points, but they should be cleaned up in a revision.
[Box 2.6] The box calls the distance function a 'well defined metric' after listing only positivity and zero-on-equality. A metric also requires symmetry and the triangle inequality. The statement is mathematically imprecise.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the notes are a tutorial that reproduces standard methods from external references; typos and internal inconsistencies do not constitute circular reasoning.

full rationale

The manuscript is a set of lecture notes, not an original derivation with predictions. Each technique (backpropagation, CNNs, VAEs, GANs, RBMs, normalising flows, Ising classification) is introduced from standard definitions and trained on data in the usual way. The Ising critical-temperature estimate is obtained by training a CNN on configurations labelled by phase (ordered vs disordered) while excluding the critical region, then locating the confidence minimum; the estimated Tc is a derived quantity and is not used as a training label, so the result does not reduce to its inputs by construction. The RBM partition function at Eq. (3.24) is misprinted as Z = sum_z E(z) instead of sum_z exp(-E(z)), and the subsequent derivation (Eqs. 3.36-3.39) uses the correct exponential form; this is an internal inconsistency/correctness issue, not a circular step. Likewise, the t-SNE KL-divergence remark and Eq. (2.5) dimension/bias-index mismatches are errata that undermine the pedagogical promise but do not make any claim equivalent to its own premise. The notes cite external literature for standard results; no load-bearing claim is justified solely by a self-citation. Therefore the circularity score is 0.

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

For a review, the central claim rests on standard background results in machine learning and statistical mechanics. The only substantive axiom-level issue found is the sign error in the KL divergence identity, which contradicts the paper's own definition of Shannon entropy. No free parameters are fitted for a new scientific claim; the numbers in the worked examples are standard hyperparameters.

assumptions (4)

standard math Universal Approximation Theorem: a sufficiently wide or deep neural network can approximate any continuous function on a compact domain
Invoked in Sec. 2.1.2 as the theoretical basis for neural network expressivity; accepted as background and not proved in the notes.
domain assumption Onsager's exact solution of the 2D Ising model: Tc = 2/ln(1+sqrt(2)) and exact magnetization/energy formulas
Used in Sec. 2.5 as ground truth for the neural-network estimate; the solution is cited from literature, not derived.
standard math Shannon entropy convention H = -sum p log p
Invoked in Sec. 2.2.1 and Box 2.8; the notes' own equations (2.19) and (2.24) are inconsistent with this convention, writing KL = H + Lce instead of KL = -H + Lce.
domain assumption Manifold hypothesis: important features of high-dimensional data lie on a lower-dimensional manifold
Stated in Sec. 3.1 as the justification for unsupervised dimensionality reduction; a heuristic rather than a proven theorem.

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Cite this review

Pith. "Pith review of Deep Learning in Classical and Quantum Physics." pith.science (2026). https://pith.science/paper/FEZABY5P

@misc{pith2026250810666,
  author       = {Pith},
  title        = {Pith review of: Deep Learning in Classical and Quantum Physics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FEZABY5P}},
  note         = {Machine review of arXiv:2508.10666}
}

read the original abstract

Scientific progress is tightly coupled to the emergence of new research tools. Today, machine learning (ML)-especially deep learning (DL)-has become a transformative instrument for quantum science and technology. Owing to the intrinsic complexity of quantum systems, DL enables efficient exploration of large parameter spaces, extraction of patterns from experimental data, and data-driven guidance for research directions. These capabilities already support tasks such as refining quantum control protocols and accelerating the discovery of materials with targeted quantum properties, making ML/DL literacy an essential skill for the next generation of quantum scientists. At the same time, DL's power brings risks: models can overfit noisy data, obscure causal structure, and yield results with limited physical interpretability. Recognizing these limitations and deploying mitigation strategies is crucial for scientific rigor. These lecture notes provide a comprehensive, graduate-level introduction to DL for quantum applications, combining conceptual exposition with hands-on examples. Organized as a progressive sequence, they aim to equip readers to decide when and how to apply DL effectively, to understand its practical constraints, and to adapt AI methods responsibly to problems across quantum physics, chemistry, and engineering.

Figures

Figures reproduced from arXiv: 2508.10666 by the authors.

**Figure 1.2.** Langton’s Loop, similarly to Conway’s Game of Life, highlights how simple rules can lead to the appearance of order and complexity, a principle applicable to fields such as robotics, autonomous systems, and artificial intelligence. It serves as a metaphor for understanding self-organization in biological systems and demonstrates how localized interactions can give rise to macroscopic patterns without centralized c… view at source ↗

**Figure 1.1.** Time-lapse of Conway’s Game of Life “glider", first discovered by Richard K. Guy in 1969. The animation begins with the canonical five-cell seed ( t = 0) and advances one update per frame. The pattern repeats every four steps while translating one cell down and one cell right, illustrating both the glider’s period-4 oscillator nature and its ability to propagate information diagonally across the lattice. The sequenc… view at source ↗

Figures from the paper (61 more)

**Figure 1.3.** Figure 1.3: Landscape of artificial intelligence (AI). Nowadays, the most im [PITH_FULL_IMAGE:figures/full_fig_p013_1_3.png]

**Figure 2.1.** Figure 2.1: Pictorial representation of a neural network. For a precise definition [PITH_FULL_IMAGE:figures/full_fig_p024_2_1.png]

**Figure 2.2.** Figure 2.2: Graphical representation of a non-linear function [PITH_FULL_IMAGE:figures/full_fig_p027_2_2.png]

**Figure 2.3.** Figure 2.3: Approximation of function g(x) = x 2 sin(πx) − 1 2+cos(x) with neural network ansatz, Eq.2.9, with a single hidden layer with N = 2, 3, 4 neurons, schematically presented on the bottom left panel. . which gets its name from the fact that it calculates the arithmetic …

**Figure 2.4.** Figure 2.4: Three steps of gradient descent visualised for a two-component [PITH_FULL_IMAGE:figures/full_fig_p036_2_4.png]

**Figure 2.5.** Figure 2.5: Visual representation of neural network training. Data, [PITH_FULL_IMAGE:figures/full_fig_p037_2_5.png]

**Figure 2.6.** Figure 2.6: Examples of hand-written digits from the MNIST dataset. Each [PITH_FULL_IMAGE:figures/full_fig_p038_2_6.png]

**Figure 2.7.** Figure 2.7: Performance of the simple NN architecture for MNIST dataset [PITH_FULL_IMAGE:figures/full_fig_p042_2_7.png]

**Figure 2.8.** Figure 2.8: Performance of the simple NN architecture for MNIST dataset [PITH_FULL_IMAGE:figures/full_fig_p043_2_8.png]

**Figure 2.9.** Figure 2.9: Examples of images of 10 classes from CIFAR-10 dataset [PITH_FULL_IMAGE:figures/full_fig_p044_2_9.png]

**Figure 2.10.** Figure 2.10: Simple neural network performance on CIFAR-10 dataset. Neural [PITH_FULL_IMAGE:figures/full_fig_p044_2_10.png]

**Figure 2.** Figure 2: Fig.2.10 presents change of loss function during training and confusion ma [PITH_FULL_IMAGE:figures/full_fig_p044_2.png]

**Figure 2.11.** Figure 2.11: Example of convolution operation, Eq.(2.37). Kernel of size [PITH_FULL_IMAGE:figures/full_fig_p045_2_11.png]

**Figure 2.12.** Figure 2.12: Action of Sobel filters for horizontal and vertical edge detection [PITH_FULL_IMAGE:figures/full_fig_p046_2_12.png]

**Figure 2.13.** Figure 2.13: Simple Convolutional Neural Network Simple neural network per [PITH_FULL_IMAGE:figures/full_fig_p047_2_13.png]

**Figure 2.** Figure 2: Fig.2.13 presents training and inference of simple CNN architecture. Neural [PITH_FULL_IMAGE:figures/full_fig_p047_2.png]

**Figure 2.14.** Figure 2.14: Convolutional Neural Network performance on CIFAR-10 dataset [PITH_FULL_IMAGE:figures/full_fig_p049_2_14.png]

**Figure 2.15.** Figure 2.15: Regularized Convolutional Neural Network performance on [PITH_FULL_IMAGE:figures/full_fig_p051_2_15.png]

**Figure 2.16.** Figure 2.16: Computational graph representing two dimensional function [PITH_FULL_IMAGE:figures/full_fig_p054_2_16.png]

**Figure 2.17.** Figure 2.17: Visual comparison between two gradient descent strategies. Steps [PITH_FULL_IMAGE:figures/full_fig_p058_2_17.png]

**Figure 2.18.** Figure 2.18: Mean magnetization pers spin, and mean energy per spin, averaged [PITH_FULL_IMAGE:figures/full_fig_p064_2_18.png]

**Figure 2.19.** Figure 2.19: Example of spins configurations on square lattice [PITH_FULL_IMAGE:figures/full_fig_p069_2_19.png]

**Figure 2.20.** Figure 2.20: Class prediction confidence vs temperature [PITH_FULL_IMAGE:figures/full_fig_p071_2_20.png]

**Figure 2.21.** Figure 2.21: Inside a sealed Chinese Room, a person who speaks no Chinese [PITH_FULL_IMAGE:figures/full_fig_p073_2_21.png]

**Figure 3.1.** Figure 3.1: Visual representation of Principle Component Analysis (PCA) di [PITH_FULL_IMAGE:figures/full_fig_p077_3_1.png]

**Figure 3.2.** Figure 3.2: The Auto Encoder (AE) architecture of three components (left [PITH_FULL_IMAGE:figures/full_fig_p081_3_2.png]

**Figure 3.3.** Figure 3.3: An example architecture for a Variational Auto-Encoder (VAE). [PITH_FULL_IMAGE:figures/full_fig_p085_3_3.png]

**Figure 3.4.** Figure 3.4: An example visualisation for mode collapse. In both plots, the [PITH_FULL_IMAGE:figures/full_fig_p089_3_4.png]

**Figure 3.5.** Figure 3.5: Two types of Boltzmann machine architectures. In the restricted [PITH_FULL_IMAGE:figures/full_fig_p091_3_5.png]

**Figure 3.6.** Figure 3.6: Comparison between the target density p ∗ (filled histogram) and the learned RBM model Pθ (red step) on [0, 1]. The distasnce between probability distributions can be quantified with the Kullback-Leibrer divergenec DKL(p ∗ ∥ Pθ) = 0.04 computed on the same discretiz…

**Figure 3.7.** Figure 3.7: Visualisation of a normalising flow model over two-variable density [PITH_FULL_IMAGE:figures/full_fig_p103_3_7.png]

**Figure 3.8.** Figure 3.8: Comparison of true trajectory xT (t) (black line) with vanilla NN (red, thick-dashed line) and PINN (blue dotted line). Times shown in this figure are in the training domain, whilst [PITH_FULL_IMAGE:figures/full_fig_p105_3_8.png]

**Figure 3.9.** Figure 3.9: (Left) Phase Space trajectories of the vanilla model (red dashed) [PITH_FULL_IMAGE:figures/full_fig_p106_3_9.png]

**Figure 3.10.** Figure 3.10: Residual plots of the vanilla NN and its PINN counterpart at [PITH_FULL_IMAGE:figures/full_fig_p107_3_10.png]

**Figure 4.1.** Figure 4.1: A unitary gate Uˆ applied to state |q⟩, resulting in |q ′ ⟩. In classical computing, we rely on a small set of logical gates such as AND, OR, and NOT, to perform complex computations. Remarkably, any Boolean function can be implemented using just a single type of gat…

**Figure 4.2.** Figure 4.2: The Hadamard and NOT gate circuits with matrix form in the computational basis Rˆ j (θ) = e i θ 2 σˆj [PITH_FULL_IMAGE:figures/full_fig_p114_4_2.png]

**Figure 4.4.** Figure 4.4: Bloch sphere representation of a single qubit pure state (red arrow). [PITH_FULL_IMAGE:figures/full_fig_p115_4_4.png]

**Figure 4.5.** Figure 4.5: The CNOT, SWAP and C-U gates and their matrix form in the computational basis. As its name suggests, the SWAP gate swaps the two qubits it acts on. The C-U gate applies the one-qubit unitary U if and only if the control qubit is in the state |1⟩. |q1⟩ |q2⟩ |q3⟩ Uˆ 3…

**Figure 4.6.** Figure 4.6: A three-qubit unitary gate in quantum circuit notation. [PITH_FULL_IMAGE:figures/full_fig_p116_4_6.png]

**Figure 4.7.** Figure 4.7: Quantum circuit to create a Bell state |Φ +⟩. First a Hadamard is applied to qubit 1, then a CNOT is applied to qubit 2. In Dirac notation, this corresponds to |Φ +⟩ = (CNOT12)(H ⊗ 1)|00⟩. decomposition in the eigenbasis of the Zˆ = PL i=1 Zˆ i operator, Zˆ|vl⟩ = vl …

**Figure 5.1.** Figure 5.1: PCA analysis of maximal representative of 3-qubit entanglement [PITH_FULL_IMAGE:figures/full_fig_p142_5_1.png]

**Figure 5.2.** Figure 5.2: Non-linear dinmensionality reduction via tSNE show distint [PITH_FULL_IMAGE:figures/full_fig_p143_5_2.png]

**Figure 5.3.** Figure 5.3: Density plot of bipartite entanglement von Neuman entropy [PITH_FULL_IMAGE:figures/full_fig_p145_5_3.png]

**Figure 5.4.** Figure 5.4: Dimensionality reduction of the ground state Cholesky vector [PITH_FULL_IMAGE:figures/full_fig_p146_5_4.png]

**Figure 5.5.** Figure 5.5: Phase boundaries detection with anomaly detection. We choose [PITH_FULL_IMAGE:figures/full_fig_p147_5_5.png]

**Figure 5.6.** Figure 5.6: An N-qubit parametric quantum circuit realised by applying an N-qubit Unitary gate to |0⟩ ⊗N . Our specific choice of decomposition for U(θ) constitutes a circuit ansätze. There is no best ansätze in general, and a specific circuit choice is often informed by the str…

**Figure 5.7.** Figure 5.7: Four example block structures for an N-qubit variational circuit with (a) and (b) based on Tensor Networks and (c) and (d) based on alternating layers. Each white box corresponds to a Unitary transform over the number of wires it connects. For example, a white box co…

**Figure 5.8.** Figure 5.8: Three different block decompositions. 5.8a has a bi-partite layered [PITH_FULL_IMAGE:figures/full_fig_p151_5_8.png]

**Figure 5.9.** Figure 5.9: Parametrized Quantum Circuit used for Variational Quantum Eigen [PITH_FULL_IMAGE:figures/full_fig_p156_5_9.png]

**Figure 5.10.** Figure 5.10: Variational energy convergence vs epochs for [PITH_FULL_IMAGE:figures/full_fig_p156_5_10.png]

**Figure 5.11.** Figure 5.11: Visualisation of QUBO problem for a small (five-component) sys [PITH_FULL_IMAGE:figures/full_fig_p158_5_11.png]

**Figure 5.12.** Figure 5.12: Visualization of the MaxCut problem. The graph nodes are parti [PITH_FULL_IMAGE:figures/full_fig_p159_5_12.png]

**Figure 5.13.** Figure 5.13: An example NQS architecture based on a multi-layer perceptron, [PITH_FULL_IMAGE:figures/full_fig_p162_5_13.png]

**Figure 5.14.** Figure 5.14: Visualisation of N qubit quadratic Hamiltonians with periodic boundary conditions. Each node represents a local subspace H2, with edges showing a quadratic Pauli term in the Hamiltonian. Red edges correspond to Yˆ operators, green to Zˆ and blue to Xˆ respectively, …

**Figure 5.15.** Figure 5.15: The simplest dynamical Hamiltonian Learning problem referred to [PITH_FULL_IMAGE:figures/full_fig_p172_5_15.png]

**Figure 5.16.** Figure 5.16: Architectures of the models used to simulate the time evolution of [PITH_FULL_IMAGE:figures/full_fig_p178_5_16.png]

**Figure 5.17.** Figure 5.17: Comparison of convergence to ground truth parameters for a six [PITH_FULL_IMAGE:figures/full_fig_p180_5_17.png]

**Figure 5.18.** Figure 5.18: Loss function (K-L divergence) from Eq. (5.97) plotted against [PITH_FULL_IMAGE:figures/full_fig_p186_5_18.png]

**Figure 5.19.** Figure 5.19: Marginal distribution of the ground-truth Q-function (left) and the [PITH_FULL_IMAGE:figures/full_fig_p187_5_19.png]

**Figure 5.20.** Figure 5.20: Samples from the conditional distribution of the ground-truth [PITH_FULL_IMAGE:figures/full_fig_p188_5_20.png]

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