PaperlayerView exact revision on arXiv

Source-linked AI summary

Quantum Machine Learning for Chemistry and Physics

Manas Sajjan, Junxu Li, Raja Selvarajan, Shree Hari Sureshbabu, Sumit Suresh Kale, Rishabh Gupta, Vinit Singh, Sabre Kais

arXiv:2111.00851v2physics.chem-phcond-mat.mtrl-sciquant-ph

TL;DR

Machine learning has been applied to physico-chemical problems, but existing reviews provide limited integration of its foundational methods and domain-specific examples. This review surveys classical and quantum-computing-enhanced approaches, their applications, and statistical-physics perspectives on learning. It synthesizes work across several application areas while noting constraints in particular approaches such as QM/MM.

  • Problem

    A review combining machine-learning fundamentals with domain-specific physico-chemical examples and applications is lacking.

    Source: §1.1 · p. 3

  • Method

    The review surveys classical machine-learning methods, quantum-computing-enhanced variants, physico-chemical applications, and statistical-physics tools for examining learning mechanisms.

    Source: §1.1 · p. 3, §7 · p. 82

  • Results

    The review covers applications including state tomography, classification, electronic structure, force fields, and drug discovery, alongside statistical-physics perspectives on learning mechanisms.

    Source: §7 · p. 82

  • Takeaways & Limitations

    The review consolidates machine-learning tools and physico-chemical applications for researchers to use synergistically.

    Source: §1.1 · p. 3

    QM/MM may require a large reactive region to reach converged results, increasing computational cost.

    Source: §5.4.1 · p. 65

Abstract

from arXiv · show

Machine learning (ML) has emerged into formidable force for identifying hidden but pertinent patterns within a given data set with the objective of subsequent generation of automated predictive behavior. In the recent years, it is safe to conclude that ML and its close cousin deep learning (DL) have ushered unprecedented developments in all areas of physical sciences especially chemistry. Not only the classical variants of ML , even those trainable on near-term quantum hardwares have been developed with promising outcomes. Such algorithms have revolutionzed material design and performance of photo-voltaics, electronic structure calculations of ground and excited states of correlated matter, computation of force-fields and potential energy surfaces informing chemical reaction dynamics, reactivity inspired rational strategies of drug designing and even classification of phases of matter with accurate identification of emergent criticality. In this review we shall explicate a subset of such topics and delineate the contributions made by both classical and quantum computing enhanced machine learning algorithms over the past few years. We shall not only present a brief overview of the well-known techniques but also highlight their learning strategies using statistical physical insight. The objective of the review is to not only to foster exposition to the aforesaid techniques but also to empower and promote cross-pollination among future-research in all areas of chemistry which can benefit from ML and in turn can potentially accelerate the growth of such algorithms.

1 Introduction

Machine learning has found applications across the physical sciences, including chemistry, while quantum computing raises questions about whether it can expand machine learning’s capabilities. This review combines classical and quantum-computing-enhanced methods, domain applications, and statistical-physics perspectives on learning.

1.1 Scope of the Review

  • The review addresses a gap in resources that combine machine-learning fundamentals with domain-specific physico-chemical examples and applications.It also aims to clarify learning mechanisms using statistical-physics tools.Source: §1.1 · pp. 3–4
  • It covers classical machine-learning tools and their quantum-computing analogues, discussing their frameworks and applications separately.Applications involving molecular, atomic, or nanomaterial data are called quantum machine learning even when analyzed classically; methods trained using quantum computers are called quantum computing enhanced.Source: §1.1 · p. 3
  • Five application domains are surveyed, with most examples using classical machine learning on quantum data because quantum-computing variants remain nascent.The listed domains are state tomography, state and phase classification, electronic structure, force-field parameterization, and drug discovery; the list is not exhaustive.Source: §1.1 · p. 3

1.2 Organization of the Review

  • The review is organized into quantum-computing foundations, machine-learning architectures, quantum-computing advantages, applications, statistical-physics accounts of learning, and emerging areas.Section 2 supplies foundations; Section 3 covers architectures; Sections 4–7 cover quantum advantages, applications, learning mechanisms, and emerging domains.Source: §1.2 · p. 4

2 Short Primer on Quantum Computing

The primer introduces qubits and quantum states, then explains gates, circuits, and quantum annealing as operational foundations for the review’s later material. It also distinguishes near-term noisy devices from fault-tolerant computing.

2 Short Primer on Quantum Computing

  • A qubit uses the computational basis states |0⟩ and |1⟩, which may correspond to distinct physical states in platforms such as trapped ions or superconducting circuits.The two states form a basis for a two-dimensional state space.Source: §2 · p. 4

2.1 Single qubit state

  • Single-qubit states can be represented on the Bloch sphere; pure states lie on its surface, while mixed states may lie inside.The angles θ and φ parameterize the states on the surface, and projective measurement collapses a superposition into a basis state with specified probabilities.Source: §2.1 · pp. 4–5, Figure 1 · p. 4

2.2 Multi-qubit state

  • Multi-qubit systems have 2^N computational-basis states, and states that do not factor into single-qubit states are entangled.Measurements yield basis states with probabilities given by squared coefficient magnitudes; multi-qubit states can represent interacting systems in quantum simulation.Source: §2.2 · pp. 5–6

2.3 Quantum gates and Quantum circuit based paradigm

  • Quantum gates are unitary and reversible; measurements introduce irreversibility, while single- and multi-qubit gates can create superpositions and correlations.Universal gate sets can approximate arbitrary unitaries, but universality does not guarantee an efficient-length circuit; NISQ algorithms generally use shallow, often variational circuits and classical processing.Source: §2.3 · pp. 6, 7, 9

2.4 Quantum Annealing based paradigm

  • Quantum annealing encodes an optimization solution in the target Hamiltonian’s ground state and slowly evolves from an easily prepared initial ground state.The review says this paradigm will receive little coverage beyond a small number of examples.Source: §2.4 · p. 9

3.1 Overview

Machine-learning problems are grouped into supervised, unsupervised, and reinforcement learning according to their data and learning settings. The section outlines each category and notes that methods can be combined.

3.1.1 Supervised learning

  • Supervised learning trains a hypothesis on labeled examples by minimizing empirical loss, then evaluates generalization on held-out test data.Classification uses discrete labels, while regression uses continuous labels; training also depends on manually chosen hyperparameters.Source: §3.1.1 · p. 9

3.1.2 Unsupervised learning

  • Unsupervised learning seeks structure in unlabeled data through tasks such as clustering, distribution learning, and dimensionality reduction.Training can minimize a divergence between the data distribution and the model distribution.Source: §3.1.2 · pp. 9–10
  • Unsupervised processing can precede supervised classification in hybrid or semi-supervised workflows.Examples include dimensionality reduction, noise filtering, or distribution learning before labels are introduced.Source: §3.1.2 · p. 10

3.1.3 Reinforcement learning

  • Reinforcement learning trains an agent through environment interaction and actions to maximize a reward, with optimal control theory playing an important role.The setting is suited to learning through interaction constrained by a finite set of actions.Source: §3.1.3 · p. 10

3.2 Classical and Quantum variants of commonly used algorithms

The section reviews classical and quantum-enhanced machine-learning methods, explaining their theoretical frameworks and how quantum variants implement familiar algorithms. It covers kernel methods, ridge regression, and principal component analysis.

3.2.1 Kernel based learning theory

  • Kernel methods avoid explicitly constructing high-dimensional feature maps by computing inner products directly, supporting regression, classification, and dimensionality-reduction algorithms.For SVM classification, mapping data into a higher-dimensional feature space can make it linearly separable.Source: §3.2.1 · p. 10

3.2.1.1 Quantum enhanced variants

  • Quantum kernels encode each input as a quantum state and use the trace inner product as the kernel; their positive-semidefinite Gram matrix defines an RKHS for hypothesis functions.The review distinguishes an implicit approach, where a quantum processor estimates kernel entries for classical learning, from an explicit approach that performs learning on a quantum computer using parameterized unitaries.Source: §3.2.1.1 · pp. 10–11

3.2.2 Ridge Regression (RR)-Linear and Kernel based

  • Ridge regression fits a linear or kernel-based model by minimizing mean-squared error, with regularization helping prevent overfitting and facilitate inversion when design-matrix columns are dependent.Kernel ridge regression uses trainable coefficients and kernel hyperparameters, which can be fixed initially and tuned across separate runs.Source: §3.2.2 · p. 11
  • Quantum linear-regression methods encode fitted values in quantum states; some permit efficient prediction without extracting exact parameters, while variational methods optimize a candidate solution through a cost function.The cited variational method’s global cost suffers from barren plateaus near a 250 × 250 design matrix; local merit functions are introduced to address trainability.Source: §3.2.2 · p. 12
  • The review reports a variational linear-equation algorithm implemented on 210 × 210 data, with logarithmic error-tolerance dependence, near-linear condition-number dependence, and poly-logarithmic qubit requirements.The review also describes a quantum ridge-regression algorithm with a ridge-parameter choice procedure resembling K-fold cross-validation, and notes no direct quantum kernel-ridge algorithm had been proposed.Source: §3.2.2 · p. 12

3.2.3 Principal Component Analysis-Linear and Kernel based

  • Linear PCA projects mean-centered data onto eigenvectors of its covariance matrix, retaining the R directions with largest eigenvalues; kernel PCA obtains projections through the Gram matrix without explicitly constructing the feature map.The target representation Z has R dimensions, with R ≤ d.Source: §3.2.3 · pp. 12–13

3.2.3.1 Quantum enhanced variants

  • Quantum PCA uses phase estimation on a positive-semidefinite matrix encoded as a reduced density matrix to obtain principal eigenvalues and eigenvectors; the cited method has time complexity O(Rlog(m)).The review says its high-qubit and gate requirements make this phase-estimation procedure cost-prohibitive for near-term devices.Source: §3.2.3.1 · p. 13
  • A single-ancilla quantum PCA method estimates eigenvalues by sweeping the probe frequency for resonance, while a direct kernel-based approach prepares Gram-matrix eigenvector columns scaled by square roots of their eigenvalues.The single-ancilla method was experimentally implemented on a two-qubit nitrogen-vacancy-center system using dynamical decoupling; the direct kernel-based method uses quantum phase estimation and controlled rotations.Source: Figure 4 · p. 13, §3.2.3.1 · pp. 13–14

3.2.4 k-nearest neighbors algorithm (k-NN)

This section surveys classical and quantum machine-learning methods for representing and classifying data, including quantum kernel approaches that encode classical data into quantum states. It also outlines proposed computational advantages and conditions or limitations relevant to their use.

3.2.4 k-nearest neighbors algorithm (k-NN)

  • k-NN assigns a query the majority label among its k nearest training instances, with distance metrics, learned metrics, centroid reduction, and voting weights shaping classification [152–155].The example uses k=5 and Euclidean distance, yielding a red-triangle label from four red neighbors and one blue neighbor.Source: §3.2.4 · p. 14

3.2.4.1 Quantum enhanced variants

  • Quantum nearest-neighbor methods encode test-to-training distances in amplitudes, amplify them coherently, and predict the nearest instance’s label [56]; quantum distance estimation was reported as O(logN) versus O(N) classically [157].Quantum k-NN variants with heterogeneous distance metrics have also been proposed for pattern-recognition problems [153, 158, 159].Source: §3.2.4.1 · p. 15

3.2.5 Decision Trees

  • Decision trees recursively partition data by feature values; candidate split criteria include information gain and Gini index, while finding optimal binary trees is NP-complete [160–163].The example classifies substances first by state at STP and, for solids, by electrical resistivity at STP.Source: §3.2.5 · p. 15
  • Quantum decision trees require criteria such as quantum entropy impurity because superposition complicates classical node-splitting rules; proposed approaches include a C5.0-based construction [165–167].Farhi and coworkers’ design can be implemented on a quantum computer with enough spin-1/2 particles; some trees were penetrated exponentially faster than by a classical random walk, though other classical algorithms solve them in polynomial time [164].Source: §3.2.5 · p. 16

3.2.6 Bayesian networks (BN)

  • A Bayesian network represents causal relationships as a directed acyclic graph of variables, with conditional probability tables; learning generally entails constructing the graph and estimating parameters [168, 169].The scattering example treats initial states as causes and observed results as effects, assuming binary features.Source: §3.2.6 · p. 16

3.2.6.1 Quantum enhanced variants

  • Quantum BN proposals replace probabilities with amplitudes or quantum probabilities [170–172], while circuit-based representations of discrete BNs reportedly achieve a square-root speedup per sample through quantum rejection sampling [173].Circuit construction maps nodes to qubits, probabilities to amplitudes, and uses single-qubit and controlled rotations; generic discrete nodes with two or more states have also been represented [174].Source: §3.2.6.1 · pp. 16, 18

3.2.7 Support vector machine (SVM)

  • SVMs map data into a feature space and construct a maximum-margin separating hyperplane; soft margins allow misclassifications, kernels handle nonseparable data, and multiclass tasks require multiple binary problems [175].The SVM optimization reaches a global minimum rather than a local one.Source: §3.2.7 · p. 18

3.2.7.1 Quantum enhanced variants

  • Quantum SVM uses matrix inversion to obtain hyperplane parameters, a training-data oracle to prepare the data state, and U(x′) to encode a test instance; solving linear equations is reported to yield exponential speedup [177, 178].For the described binary classification, the test label is positive when the final-state expectation value is greater than zero and negative otherwise [177].Source: §3.2.7.1 · pp. 18–19
  • In 2019, Xanadu researchers proposed estimating quantum-state inner products to compute a classically intractable kernel for use in classical methods such as SVM [179].The kernel can be fed into any classical kernel method.Source: §3.2.7.1 · p. 19
  • Quantum feature maps nonlinearly encode a classical data point x as an n-qubit state |φ(x)⟩ = U(x)|0⟩ using a parameterized circuit family {U(x)} [180].Source: §3.2.7.1 · p. 19
  • In quantum kernel learning, feature maps take the role of pattern recognition.Source: §3.2.7.1 · p. 19

3.2.8 Gaussian Process Regression

The section presents Gaussian process regression (GPR) through Bayesian weight-space and function-space views, then describes quantum-enhanced kernels built from coherent states and entangled feature maps.

3.2.8 Gaussian Process Regression

  • Gaussian process regression (GPR) is a non-parametric, supervised Bayesian method for inferring an unknown mapping from d-dimensional inputs to scalar outputs.Source: §3.2.8 · pp. 19–20
  • In the weight-space view, Bayesian updating yields a posterior over model parameters that weights predictions at new inputs.Inputs can be projected through basis functions to approximate the target function.Source: §3.2.8 · p. 20
  • With a chosen kernel, the model fits coefficients by minimizing a loss that includes Tikhonov regularization.Source: §3.2.8 · p. 20
  • In the function-space view, a Gaussian process specifies distributions over functions using a mean function and covariance function k(x,x′).Source: §3.2.8 · p. 20

3.2.8.1 Quantum enhanced variants

  • Otten et al.181 construct quantum-enhanced kernels by approximating the classical squared-exponential feature map with coherent states and implementing finite-dimensional displacement operators on qubits.The finite-dimensional operators are decomposed into log2(N) Pauli operators and Trotterized for m steps; Fig. 12 schematizes the quantum-enhanced kernel for GPR.Source: §3.2.8.1 · pp. 20–21, Figure 12 · p. 20
  • A multi-mode squeezing operator entangles data dimensions, with the proposed kernel intended to let smaller quantum devices perform higher-dimensional regression using few operations.The GP-based learning task is then performed on classical hardware.Source: §3.2.8.1 · p. 21

3.3 Artificial Neural networks

This section surveys neural-network architectures used in physics and chemistry, including classical training methods and proposed quantum-enhanced variants. It covers recurrent and generative models, their memory, compression, and distribution-learning strategies, and their architectures and training objectives rather than benchmark comparisons.

3.3.1 Perceptron and Feed forward-neural networks

  • A perceptron applies an affine transformation and nonlinear activation, while feed-forward networks stack such neurons and train their weights through back-propagation.Gradients can be used with gradient descent or other optimizers; cross-validation evaluates predictions, while regularization or dropout can address overfitting. Sigmoid saturation can slow learning and near-zero gradients at extreme inputs can degrade performance, motivating alternatives such as tanh.Source: §3.3.1 · pp. 21, 23

3.3.1.1 Quantum enhanced variant

  • Quantum-classical hybrid networks use quantum unitaries for linear operations and classical layers for nonlinear activations, but finding an optimal quantum activation method remains open.Other approaches include repeat-until-success measurements and periodic activation circuits; the described periodic-function circuit has complexity O(N2⌈log2 N⌉2), with polynomial speedup only under certain circumstances.Source: §3.3.1.1 · pp. 23–24

3.3.2 Convolutional Neural Network (CNN)

  • CNNs preserve spatial correlations in image inputs by extracting and condensing feature maps before feeding them to a fully connected network.Convolutional layers detect features with spatial filters, activation layers introduce nonlinearity, and pooling downsamples feature maps.Source: §3.3.2 · pp. 24–26

3.3.2.1 Quantum enhanced variant

  • A quantum CNN classifies an unknown N-qubit state against labeled training states using repeated quasi-local convolutional unitaries, measurement-conditioned pooling, and a final fully connected layer.Pooling measurements reduce circuit width, and final measurements produce the prediction.Source: §3.3.2.1 · pp. 26–27

3.3.3 Recurrent Neural networks

  • Recurrent neural networks are designed for time-ordered sequences, where retaining information about previous entries is essential.The passage identifies natural-language processing, stock-price prediction, translation, and time-series prediction as examples.Source: §3.3.3 · p. 27
  • Vanilla RNNs share weights across sequence steps and pass hidden activations forward; LSTM and GRU architectures address vanishing or exploding gradients, with LSTMs using gated memory.LSTM gates erase, update, and expose information from a memory channel.Source: §3.3.3 · pp. 27–28

3.3.3.1 Quantum enhanced variants

  • Quantum RNNs use two qubit registers: one encodes each input and produces measured outputs, while the other retains sequence memory; a hybrid quantum-classical LSTM showed good results on damped harmonic oscillator dynamics.The quantum LSTM uses parameterized circuits for its gates and classical gradient-based updates.Source: §3.3.3.1 · pp. 28, 30

3.3.4 Autoencoders

  • Autoencoders create compact input representations with an encoder bottleneck and reconstruct them with a decoder; undercomplete, sparse, denoising, and contractive variants use different code-size, regularization, or noise choices.They are trained to minimize reconstruction error; a small latent code can limit overfitting in an undercomplete autoencoder.Source: §3.3.4 · p. 30

3.3.4.1 Quantum enhanced variants

  • Quantum autoencoders encode an ensemble of states into fewer qubits, replace the remaining trash system with a reference state, then decode; training maximizes fidelity after reconstruction.The ideal encoding leaves the code and trash systems in a product state, with the trash matching the reference state.Source: §3.3.4.1 · pp. 30–31

3.3.5 Variational Encoders

  • Variational autoencoders learn latent distributions by optimizing an ELBO; Gaussian assumptions and reparameterization enable training, while a quantum variant uses a Quantum Boltzmann Machine for gradient updates.The KL-divergence term regularizes latent-space continuity and completeness.Source: §3.3.5 · p. 31

3.3.6 Generative Adversarial Network (GAN)

  • GANs train a generator and discriminator in competition until generated and true samples become indistinguishable; conditional GANs add labels to constrain generation to a chosen class.The trained discriminator is discarded, and the generator can be repurposed for transfer learning on similar input data.Source: §3.3.6 · pp. 31–32

3.3.6.1 Quantum enhanced variants

  • QGANs feed source or generated samples, labels, and scratch qubits to a discriminator; their measurement-probability cost is linear, with gradients computed by parameter-shift or unitary decomposition.The generator receives noise and a label to produce output states.Source: §3.3.6.1 · pp. 32–33

3.4 Tensor Networks

Tensor networks build global quantum states from local tensors, using limited entanglement to represent some large-Hilbert-space states with polynomial resources. The review surveys MPS, TTN, PEPS, and MERA, the DMRG algorithm, and connections between tensor networks and quantum circuits.

3.4.1 Matrix Product State (MPS)

  • MPS represents n-qubit states with fixed bond dimension χ using O(nχ^2) parameters, and the Area Law supports this form for ground states of local gapped Hamiltonians.Canonicalization fixes the MPS gauge by choosing where to absorb the Schmidt-weight matrices.Source: §3.4 · p. 33, §3.4.1 · p. 34

3.4.2 Tree Tensor Networks (TTN)

  • TTN coarse-graining stores local correlations in lower layers and long-range correlations in upper layers; isometric tensors simplify optimization and local-observable calculations.TTN-based ML-MCTDH methods have been used to compute acetonitrile vibrational eigenstates.Source: §3.4.2 · pp. 34–35

3.4.3 Projected Entangled Pair States (PEPS)

  • PEPS generalizes MPS to higher dimensions, requiring O(N^2dD^4) storage on a square lattice, while inner products scale exponentially with bond dimension D.A balanced cut crosses O(N) bonds, producing a rank-O(N) intermediate tensor during inner-product evaluation.Source: §3.4.3 · p. 35

3.4.4 Multi-scale Entanglement Renormalisation Ansatz (MERA)

  • MERA supports gapless and near-critical systems with polynomially decaying correlations, and local expectation values can be computed by contracting an operator’s causal cone.Its layered structure allows entanglement entropy to diverge logarithmically from the Area Law.Source: §3.4.4 · pp. 35–36

3.4.5 Density Matrix Renormalization Group (DMRG)

  • DMRG finds ground states of local Hamiltonians by minimizing energy within a bounded-entanglement MPS ansatz, using iterative local optimization with sweeps over the lattice.Larger optimized site groups can improve accuracy and convergence but increase computation cost.Source: §3.4.5 · p. 36

3.4.6 Quantum enhanced Tensor Networks

  • Tensor networks connect classical circuit simulation with quantum state preparation; circuit simulation cost depends exponentially on treewidth, while quantum circuits can prepare tensor-network states.The review describes classical simulation of quantum algorithms and efforts to optimize tensor networks on quantum computers.Source: §3.4.6 · pp. 36–37
  • Hybrid TTN simulations of 9×8-qubit spin systems were up to two orders more accurate than standard DMRG on MPS and imaginary-TEBD on PEPS.Certain large-bond-dimension tensor-network states can also be prepared on quantum computers despite not being efficiently classically simulable.Source: §3.4.6 · p. 37

4 Case for Quantum computing enhanced Machine Learning

This section surveys theoretical guarantees and reported demonstrations of quantum-enhanced machine learning, including function approximation, quantum-kernel classification, and quantum neural networks, while noting conditions and limitations on claimed advantages. It also presents a qubit-efficient tensor-network classifier for handwritten digits and identifies possible extensions to chemistry and other many-body applications.

4.1 Universal function approximation through supervised learning on a quantum computer.

  • Mitarai etal 306 use data-uploading encodings to claim broader function approximation than classical supervised learning and report successful transverse-Ising time-evolution fitting and non-linearly separable binary classification.The argument links classical simulation of every such function to learning quantum cellular automata, which is not achievable with polynomial resources on a classical device.Source: §4.1 · p. 37
  • Quantum circuits can approximate any square-integrable function within arbitrary preset tolerance when encoding Hamiltonians generate arbitrary frequency ranges and the trainable circuit can realize any unitary.The Fourier spectrum’s frequencies are controlled by the encoding Hamiltonians, while its coefficients depend on the trainable unitaries and measurement operator.Source: §4.1 · p. 37

4.2 Power of Kernel estimation and data-classification from Quantum computers.

  • Two-mode squeezed-state encoding with variational processing linearly separates a dataset at 100% accuracy, but the primary squeezed-state encoding is efficiently classically simulable.The authors suggest non-Gaussian elements or other classically intractable continuous-variable circuits as routes to more demanding kernel estimation.Source: §4.2 · p. 38
  • On a 5-qubit superconducting circuit, Ref324 reports explicit-classification success close to 100% with four variational layers and 100% accuracy using hardware-estimated kernels.The hardware-estimated kernel matrix agreed fairly well with the ideal one.Source: §4.2 · p. 39
  • SVM-QKE achieves 0.99 accuracy in polynomial time with probability at least 2/3 on the discrete-log task, while classical learning is bounded by 0.5+1/poly(n); quantum kernels also outperform classical kernels on complex data, but not necessarily simple, well-separated data.High noise can impair kernel estimation, while classical preprocessing such as discarding negative eigenvalues is proposed to mitigate noise.Source: §4.2 · pp. 39–40

4.3 Power of Quantum-Neural Networks

  • For d=40, the full QNN has higher effective dimension, trains faster, and reaches lower loss than the easy-quantum and classical networks, with the reported conclusions retained on real hardware.Its Fisher-information eigenvalues were also more uniformly distributed than those of the classical network; the easy-quantum model fell between the two.Source: §4.3 · pp. 40–41
  • With perfect agreement on training samples, entanglement rank r=1 requires t=d=2^n samples, whereas r=d permits one; controlling entanglement lowered average risk below the classical limit on Rigetti’s processor.For intermediate ranks, the required sample count can be far below d.Source: §4.3 · pp. 41–42
  • For 1D symmetry-protected phases, QCNN uses O(log(n)) parameters and fewer samples; its gradients vanish polynomially, while bQCNN is more expressive at similar depth but requires more parameters.The QCNN parameter requirement is described as a two-fold exponential reduction relative to other quantum classifiers.Source: §4.3 · p. 42

4.4 Power of Quantum Computers for Tensor-Network based Machine Learning tasks

  • The reported quantum advantage may extend to ligand-selectivity classification for drug design and tensor-network applications in spin liquids, excitonic materials, and molecules.The section points to further examples of these prospects in Sections 5.2, 5.3.2, and 5.5.1.Source: §4.4 · p. 42
  • A tree tensor-network ansatz encodes input data into qubits, applies parameterized coarse-graining gates, and measures output-label probabilities; sequential resets enable operation with O(V log(N)) physical qubits.For the illustrated case with two virtual qubits and 16 input states, the architecture requires 8 qubits. The optimization uses an SPSA-based procedure with momentum in its gradient estimates.Source: §4.4 · pp. 42–43, Figure 22 · p. 43
  • The classifier reported a lowest percentage error of 87% and average test accuracy over 95%, with performance degradation of only 0.004 under 10× noise parameters.The authors also report similar accuracies with a reduced bond dimension compared with a classical TTN implementation.Source: §4.4 · p. 43

5.1 State preparation protocols and Quantum State Tomography

Quantum state tomography (QST) is challenged by exponentially scaling measurement and classical-processing demands, and by noisy or inaccessible measurements. This review surveys machine-learning approaches to state preparation and reconstruction, including classical and quantum-enhanced methods.

5.1 State preparation protocols and Quantum State Tomography

  • RBM tomography reconstructed entangled many-body states from limited measurements, but performed worse on random-unitary states; demonstrations used real wavefunctions, reducing measurement overhead.The review notes open questions about training-set size, hidden-neuron count, and comparisons with other methods.Source: §5.1 · p. 45
  • A local-measurement neural network reconstructed states up to 7 qubits in simulation and 4-qubit NMR states experimentally, with MLE-comparable fidelity, faster processing, and better noise tolerance.The approach learns to infer the Hamiltonian from local measurements, addressing the computational hardness of reconstruction from reduced density matrices.Source: §5.1 · p. 45, Figure 23 · p. 46
  • A CNN mapped noisy or incomplete coincidence measurements through a predicted τ-matrix to a density matrix and improved fidelity over Stokes reconstruction in both settings.The reported comparison covers noisy data and incomplete projective measurements.Source: §5.1 · p. 45
  • Adaptive tomography methods reduce measurement needs or accelerate processing: ABQT reduced measurements for two-qubit pure states, while RAQST outperformed nonadaptive methods for high-purity two-qubit states.The review reports RAQST as an adaptive extension of recursive linear-regression estimation.Source: §5.1 · pp. 46–47
  • NAQT accelerated reconstruction by a factor of a million for approximately 10^7 measurements by replacing Bayesian updates with a neural network and avoiding costly resampling.The review reports retained accuracy and independence from qubit count and measurement type.Source: §5.1 · p. 46
  • QGAN reproduced quantum-channel-simulator data statistics at 98.8% average fidelity in a superconducting-circuit experiment, while CGAN-based QST reported higher fidelity, faster convergence, and fewer measurements than MLE.The CGAN report also states that sufficient training on simulated data can permit single-shot reconstruction.Source: §5.1 · pp. 47–48
  • A Transformer-based attention tomography method improved GHZ-state learning complexity by an order of magnitude over RNN and reconstructed a 6-qubit GHZ state at 0.977 fidelity, beyond IBM Qiskit's stated capabilities.It also achieved superior reconstruction fidelity to MLE for a 3-qubit system and was described as scalable to larger systems.Source: §5.1 · p. 48
  • Machine-learning QST spans state-preparation circuits, generative models, adaptive methods, and neural-network reconstruction, with possible roles in NISQ-device validation and benchmarking.The review also notes that loading classical data onto quantum devices can be a bottleneck.Source: §5.1 · pp. 44, 45, 48

5.2 State classification protocols

The section reviews supervised classification of physico-chemical data and unsupervised approaches that identify boundaries relevant to phase transitions. It describes classical and quantum methods, their evaluation, and examples including phase classification in VO2 and spin models.

Supervised classification workflow

  • A typical supervised classification workflow preprocesses experimental data, selects and trains a classifier, predicts test labels, and evaluates error, with cross-validation among the evaluation options [161, 467].The review notes that measurement errors and reaction impurities can leave noise and missing feature values in raw datasets.Source: §5.2 · pp. 49, 52, 53, Figure 25 · p. 49

Decision trees

  • Decision trees can expose rules in reactive-process classification through the reactant-to-barrier method, although quantum decision trees are rarely used independently for intricate chemical problems [472].The tree’s branches encode accepted or declined substituent changes, and its energy levels summarize barrier ranges.Source: §5.2 · p. 49, Figure 26 · p. 50

Bayesian networks

  • Bayesian networks have been applied to chemical-shift prediction, entropy-driven phase transitions, and quantum molecular dynamics [473, 474, 475].The review notes that Bayesian-network methods can achieve maximum prediction accuracy with relatively small datasets and train in a single pass.Source: §5.2 · pp. 49, 50, 53

Quantum instance-based learning

  • Quantum clustering followed by classification predicts insulating and metallic regions of VO2 from pressure-temperature instances [318].The figure shows training labels, cluster sublabels, and test-instance predictions.Source: §5.2 · p. 50, Figure 28 · p. 51

Quantum neural networks

  • A supervised quantum neural network framework encodes binary inputs and measures an auxiliary qubit for predictions, but simulations reported no speedup over classical competitors [485].Training minimizes a loss over labeled examples.Source: §5.2 · p. 52

Quantum classifiers

  • Quantum feature maps encode inputs in a quantum Hilbert space, while circuit-centric classifiers use compact variational circuits to learn decision boundaries [179, 487].The feature-map approach uses quantum-state inner products to evaluate kernels; circuit-centric models use few trainable parameters and small numbers of one- and two-qubit gates.Source: §5.2 · p. 52

Quantum classifier with tensor network ansatz

  • 99% accuracy for the transverse-field Ising model and 94% accuracy for the XXZ model were reported using 4- and 6-layered classifiers, respectively [490].The method prepares approximate ground states with a variational quantum eigensolver, then classifies the states using a tensor-network quantum circuit.Source: §5.2 · p. 52

Unsupervised classification

  • Unsupervised learning uses unlabeled instances to find boundaries and can support phase-transition recognition, while fewer unsupervised QML applications in chemical classification have been reported [492, 493].Reviewed clustering approaches include quantum k-means, q-means, and a hybrid algorithm implemented on a 19-qubit computer.Source: §5.2 · pp. 53–54

5.3 Many-body structure and property prediction for molecules, materials, and lattice models and spin-systems

This section surveys classical machine-learning, tensor-network, and quantum-enhanced learning approaches for predicting properties of molecules and materials and modeling many-body quantum systems. It covers applications including molecular energy prediction, material band structures, constrained eigenstates, tensor-network representations, and lattice or disordered-spin systems.

5.3.1 Machine learning techniques on a classical processor

  • Machine learning learned density functionals from exact data and predicted molecular energies; a genetic-algorithm-trained neural network also solved the He and Li Kohn-Sham equations. [312, 342]Snyder et al. used kernel-ridge regression for a one-dimensional spinless-fermion kinetic-energy functional; Caetano et al. reported agreement with other numerical procedures.Source: §5.3.1 · p. 54
  • Delta-learning corrected lower-order calculations for molecular thermochemical and correlation properties, while Gaussian-process geometry optimization performed better with a Matérn than a squared-exponential kernel. [510, 513]The geometry optimizer was tested on 25 cases and a higher-dimensional molybdenum complex.Source: §5.3.1 · pp. 54–55
  • Restricted Boltzmann machine wavefunctions modeled spin-system ground states and dynamics with high accuracy, while molecular RBM ansätze outperformed CCSD(T) and Jastrow methods. [346, 514]The molecular approach maps Hamiltonians to Pauli strings and variationally minimizes ground-state energy using sampled configurations.Source: §5.3.1 · p. 55
  • Neural methods selected configuration-interaction states, predicted molecular-orbital properties, and captured Hubbard-model behavior and correlations across metallic and Mott-insulating phases. [520–523, 524]The follow-up configuration-interaction work computed N2, H2O, and CO potential-energy curves with near-full CI accuracy; SchNet and SchNOrd predicted energies, Hamiltonians, and overlap matrices for four molecules.Source: §5.3.1 · pp. 55–56
  • PauliNet achieved high-accuracy molecular energies, while autoregressive neural networks reproduced standard full-configuration-interaction results. [525, 532]PauliNet was tested on atoms and molecules including H2, LiH, Be, B, and H10, and on cyclobutadiene minimum and transition-state energies.Source: §5.3.1 · pp. 56–57
  • ML models screen photovoltaic and superconducting materials, while generative, transfer-learning, and graph models support oxide and zeolite discovery, intermetallic prediction, and mechanical-property estimation. [529, 531, 533–537]The reported applications include identifying 242 oxide compositions for NCS structures, generating 121 zeolite materials, and predicting bulk and shear moduli.Source: §5.3.1 · pp. 57–58
  • A projected-entangled-simplex-state calculation of Ca10Cr7O28 found trends in energy, heat capacity, magnetization, and susceptibility consistent with a gapless spin-liquid spectrum. [541]The modeled quantum magnet has a distorted bilayer Kagome lattice with competing ferromagnetic and antiferromagnetic interactions.Source: §5.3.1 · p. 58
  • Tensor-network methods extend simulations to higher dimensions through tree-network and fermionic-operator approaches, and to finite-temperature 2D Ising and Bose-Hubbard systems using PEPS. [542, 543, 547]The tree tensor network was applied to the 2D Heisenberg model, square-lattice spinless fermions, and Be; the fermionic-operator scheme manages sign factors during contraction.Source: §5.3.1 · pp. 58–59
  • For many-body localization, an infinite-PEPS algorithm simulated time evolution of a disordered spin-1/2 XXZ model on a square lattice, with local particle-number expectation values used to identify phases across disorder conditions.The system was initialized in the Neel state.Source: §5.3.1 · p. 59

5.3.2 Quantum-computing enhanced machine learning techniques

  • A hybrid neural network with a parameterized quantum-circuit layer and measurement-based classical nonlinearities was trained unsupervised on selected bond lengths and predicted H2, LiH, and BeH2 ground-state energies with high accuracy (206).The circuit encodes input data as quantum states, optimizes the output state, and extracts outputs as measurement expectation values.Source: §5.3.2 · p. 59
  • On two 27-qubit IBM-Q devices, RBM calculations obtained h-BN and graphene valence bands and Hubbard-U band splitting; mitigation and warm starts supported accurate TMDC frontier bands and spin-orbit splitting (Ref519; 347).For graphene, the study also considered Hubbard interactions of 0 eV and 9.3 eV; MoS2 valence and conduction bands were obtained with the constrained-minimization procedure.Source: §5.3.2 · p. 60, Figure 33 · p. 61
  • Constrained optimization can target arbitrary molecular or material energy eigenstates, filter states by Hamiltonian symmetry, and obtain excited-state and ground-state results for multireference molecular systems (347).The study explored energy errors, target-state infidelity, and constraint violation; MoS2 symmetry filtering used the squared-orbital angular momentum operator.Source: §5.3.2 · p. 60, Figure 33 · p. 61
  • Quantum-circuit tensor-network ansätze achieved accuracy comparable to standard classical variants with lower parameter cost, while relative ground-state energy error decreased inversely with a polynomial of parameter count (549).The comparisons used Heisenberg and Fermi-Hubbard models and included qMPS and qMERA circuit tensor networks.Source: §5.3.2 · p. 62, Figure 35 · p. 63
  • A Schmidt-decomposition scheme split a 2N-qubit circuit into N-qubit circuits and produced excellent water-molecule energies for stretched O–H geometries using five processor qubits and three Schmidt coefficients (552).The setup used a frozen-core approximation and U = V to enforce spin symmetry; results degraded for H–O–H bond-angle deformations.Source: §5.3.2 · pp. 62, 64
  • A tensor-network variational algorithm estimates eigenvectors of a black-box unitary by increasing MPS bond rank until the desired accuracy is reached, and terminates when the state reaches an eigenstate.Its stated implementation complexity, measured in CNOT gates, scales as O(l·n·r^2).Source: §5.3.2 · p. 64

5.4 Estimation and parameterization of force fields in molecular dynamics

The section reviews classical and quantum-computing approaches to force fields and molecular dynamics, including ML-assisted ReaxFF parameterization and quantum algorithms for molecular simulation. It describes accuracy, computational-cost, and data constraints alongside reported applications and results.

5.4.1 Machine learning techniques on a classical processor

  • Because conventional force fields omit effects such as many-body interactions, polarization, and bond breaking, their assumptions can limit physical accuracy; QM/MM also raises costs when convergence requires a large reactive region.ML force fields aim to combine ab initio accuracy with classical force-field efficiency, but costly training data make data efficiency important.Source: §5.4.1 · pp. 64–65
  • ReaxFF parameter optimization combines random-forest feature selection, k-nearest-neighbor identification of probable local minima, and grid-search optimization.Grid search groups important features and samples selected parameter sets; a cutoff can limit work on less-sensitive groups.Source: §5.4.1 · pp. 65–66
  • Optimized ReaxFF parameters predicted the α-Al2O3 crystal structure at 2000 K, while different optimized sets could diverge in their predictions of other physical properties.The review also describes gradient-based optimization using automatic differentiation, whose local optimizer efficiency is reported to be independent of the initial ReaxFF approximation.Source: §5.4.1 · p. 66
  • For metal-halide perovskites, mutual-information analysis identified geometric features as more important than atomic velocities for predicting nonadiabatic coupling.For CsPbI3, feature selection reduced a 360-feature model to 12 features while retaining high-quality NA-MD results.Source: §5.4.1 · p. 67

5.4.2 Quantum-computing enhanced machine learning techniques

  • A coarse-grained quantum-harmonic-oscillator model represents polarizable molecular charge and incorporates dispersion interactions, with measurement-circuit counts scaling as O(n) rather than O(n^3) for an orbital-based VQE Hamiltonian.The approach was used to calculate London dispersion energies for interacting I2 dimers on an IBM quantum processor; the review presents anharmonicity as a possible extension.Source: §5.4.2 · p. 68
  • The nonadiabatic wave-packet algorithm reproduced the Marcus-model population-transfer decline in the inverted region in excellent agreement with exact propagation using an 18-qubit simulation.The dynamics were not fully implemented on a quantum processor because the circuit depth exceeded coherence times available on current devices.Source: §5.4.2 · p. 68
  • The proposed nonadiabatic algorithm encodes a d-dimensional polynomial potential energy surface using O(d log^2(N)) resources, with N denoting discrete grid points.The review says extension to higher dimensions is straightforward and identifies sufficient coherence time as a condition for the anticipated advantage.Source: §5.4.2 · pp. 68–69
  • For single molecules, a quantum-neural-network potential energy surface performed competitively with classical counterparts, and the study suggests properly designed QNNs can have larger effective dimension and fast, stable training.The review presents this as a possible quantum advantage for force-field generation, not an established general advantage.Source: §5.4.2 · p. 69

5.5 Drug-Discovery Pipeline and pharmaceutical applications

This section surveys machine-learning approaches to pharmaceutical discovery, including classical methods for ligand and target screening, drug design, and toxicity prediction, as well as emerging quantum-computing applications. It also covers industry efforts to assess and prepare for quantum computing through collaborations, infrastructure, expertise, and quantum-enabled tools.

5.5 Drug-Discovery Pipeline and pharmaceutical applications

  • CADD screening draws on molecular databases and encodes molecules and targets as descriptors, from generic features and fingerprints to graph-based and three-dimensional representations.The section notes that 1D and 2D encodings may be less sensitive to stereochemistry, while 3D descriptors can capture structural details such as dihedral angles.Source: §5.5 · p. 69

5.5.1 Structure-based drug designing protocols using classical machine learning techniques

  • CNN pose prediction outperformed AutoDock Vina and remained superior in virtual screening; LSTM models performed well across four drug-target classes.The LSTM approach outperformed traditional ML techniques and multilayer perceptrons, including with a small nuclear-receptor training sample. A CNN–BiLSTM model achieved 98.7% AUC on the Human dataset and outperformed traditional methods; it also outperformed Vina and AtomNet on DUD-E and competitive algorithms on BindingDB.Source: §5.5.1 · pp. 70–71

5.5.2 Ligand-based drug designing protocols using classical machine learning techniques

  • Deep-learning networks outperformed traditional models in 14 of 19 assays, with multitask models beating single-task models in 12 of those 14.Across 259 assays and 1.6 million compounds, multitask networks performed better than several ML models, with performance improving as task count or data volume increased. Deep networks improved mean R2 over Random Forest by 0.043 with arbitrary parameters and 0.051 with refined parameters.Source: §5.5.2 · pp. 71–72

5.5.3 Machine learning and drug-induced toxicity

  • The HLAED kNN model classified hepatotoxicity with sensitivity ≥70% and specificity ≥90%, while an epoxidation neural network outperformed logistic regression and other competitors.The kNN model used an applicability-domain threshold to flag unreliable predictions. For phospholipidosis, descriptor choice and dataset size affected training, and positively charged nitrogen centers correlated with active compounds; the epoxidation model distinguished sites of epoxidation from hydroxylation.Source: §5.5.3 · pp. 72–73

5.5.4 Power of quantum computers and quantum-computing enhanced machine learning techniques

  • DeepMind’s CASP14 prediction achieved 0.95 Å root-mean-square error from the target structure for 95% of residues; quantum SVMs attained comparable accuracy on most datasets using feature-reduction techniques.A hybrid quantum-classical approach for high-throughput virtual screening achieved good accuracy and slightly faster data-processing runtime on a quantum computer than on a classical computer.Source: §5.5.4 · p. 74
  • Pharmaceutical companies should act early by assessing quantum computing’s potential value, identifying suitable product-design problems, and evaluating timelines and available resources [688, 689].The reports warn that technology firms could enter in-silico drug discovery, while pharmaceutical companies retain experience in computational drug design [688].Source: §5.5.4 · pp. 74–75
  • Quantum-computer screening of training datasets shows sublinear runtime scaling versus classical computing, alongside ligand classification for SARS-CoV-2 and tuberculosis.The figure illustrates active/inactive ligand classifications for both diseases and compares quantum and classical screening runtimes.Source: Figure 40 · p. 75
  • Companies can address hardware and software needs through technology-provider partnerships, tools such as Qiskit and OpenFermion, and specialist drug-discovery firms developing quantum or quantum-inspired methods.Examples include Amgen’s collaborations with Amazon Braket and IBMQ, Boehringer’s work with Google QuantumAI, and firms combining classical approaches with machine learning.Source: §5.5.4 · p. 75
  • A survey found about 50 quantum-technology employees across 21 pharmaceutical companies with combined 2019 revenue of C800 billion, underscoring the need for recruitment and training [687, 688, 689].The section also describes pharma-startup collaborations, including Merck investments and partnerships involving Amgen, AstraZeneca, and Cambridge Quantum Computing [690, 691, 692, 693, 694].Source: §5.5.4 · pp. 75–76

6 Insight into learning mechanisms

The section examines learning mechanisms in classical and quantum neural networks through statistical-physics perspectives, including feature extraction and learnability methods. It also addresses quantum neural-network expressibility, trainability, noise and measurement effects, generalization, and conditions that produce barren plateaus.

6 Insight into learning mechanisms

  • Statistical physics offers analytical insight into learning: replica methods study self-averaging and latent-variable estimates, while feature extraction links deep learning to renormalization-group ideas.The section frames learning through macroscopic behavior emerging from heterogeneous systems and identifies criticality and fixed points as concepts for studying learning in deep networks.Source: §6 · p. 76

6.1 Replica Method

  • For the Ising model, the replica-symmetric solution predicts q=0 for β<1 and a rising q for β>1, but its saddle point is unstable and physically inconsistent.The cavity method independently derives the self-consistency equation obtained by the replica method.Source: §6.1 · p. 78, §6.2 · p. 78

6.3 Renormalization group and RBM

  • Renormalization-group flows classify operators as relevant or irrelevant and identify critical points with phase transitions; the 2D Ising critical temperature is Tc = 2.269.The section notes that microscopic correlations are washed out under renormalization, so the invariance test applies only to observables defined on coarse-grained degrees of freedom.Source: §6.3 · p. 79
  • RBM-based renormalization methods show differing evidence of RG-like behavior: learned-weight flows approach criticality, variational RG preserves nearest-neighbor correlations, and mutual-information coarse-graining reproduces Kadanoff renormalization.The learned-weight flow predicts critical exponents [15],[16] and reproduces the long-range scaling dimension Δs with high precision, while short-range Δε has high errors. Variational RG behavior is reported for large lattices at criticality; mutual-information coarse-graining separates the phase transition and predicts the critical temperature with high accuracy.Source: §6.3.1 · p. 79, §6.3.2 · p. 80, §6.3.3 · p. 80

6.3.1 From learned weights

  • In learned-weight RBM flows, the layer width remains fixed, unlike RG, yet the flow approaches the critical point and predicts critical exponents [15],[16].A single flow reproduces the long-range scaling dimension Δs with high precision, whereas high errors in Δε indicate that short-range correlations are generally lost.Source: §6.3.1 · p. 79

6.3.2 Variational RG

  • Variational RG uses stacked RBMs to coarse-grain hidden-layer outputs; at criticality, nearest-neighbor correlations are preserved for large lattices, but no pattern appears for small lattices.The reported behavior is described as reminiscent of RG under those circumstances.Source: §6.3.2 · p. 80

6.3.3 Real Space Mutual Information

  • Real-space mutual-information coarse-graining trains RBMs to retain information between hidden spins and the environment, reproducing Kadanoff renormalization and locating the phase transition with high accuracy.The procedure repeats at each renormalization step and is demonstrated on a 128×128 square lattice.Source: §6.3.3 · p. 80

6.4 Learnability of Quantum Neural networks

  • Quantum neural networks encode classical features into quantum states and use parameterized unitaries; their ansatz expressibility relates to MNIST classification accuracy, but high expressibility risks barren plateaus.The section also frames trainability as empirical loss minimization with an L2 penalty and notes that gradients can be estimated using the parameter-shift rule.Source: §6.4 · p. 80, §6.4.1 · p. 81, §6.4.2 · p. 81

6.4.2 Trainability of QNN

  • Yuxuan et al725 bound convergence rates R1 and R2; larger K and B and lower p, d, and L improve trainability, while noise-induced gradient bias degrades learning.Because noise effects grow exponentially with circuit depth, training deep circuits may be infeasible in the NISQ era.Source: §6.4.2 · p. 81

6.4.3 Generalizability of QNN

  • Andrea Mari et al.208 demonstrate QML transfer learning across classical and quantum network combinations; a classification error bound indicates no overfitting when training pairs exceed 2^I, where I is Rényi mutual information.The transfer-learning examples cover classical-quantum, quantum-classical, and quantum-quantum networks; the generalization bound relates quantum state space and classical input space.Source: §6.4.3 · p. 81

6.4.4 Barren Plateaus in training Variational Circuits

  • Barren plateaus arise from vanishing sample-gradient variance; risks include polynomially deep 2-design ansätze, global costs, highly expressive circuits, and gate noise, which causes exponential gradient decay with qubit number and depth.For controllable systems, gradients decay with the dimension of the initial state's symmetry subspace; tailoring ansätze to remain uncontrollable can preserve approximate performance at the cost of expressibility.Source: §6.4.4 · pp. 81–82, Figure 43 · p. 82

7 Conclusions

The review surveys classical and quantum-computing-enhanced machine learning for physico-chemical applications, alongside statistical-physics perspectives on learning mechanisms. It identifies promising application areas and open challenges for quantum machine learning.

7 Conclusions

  • The review covers classical and quantum-enhanced methods across five physico-chemical application domains, while using statistical-physics tools to examine learning mechanisms.The application areas include state reconstruction and classification, electronic structure and property prediction, force-field parameterization, and drug discovery.Source: §7 · p. 82
  • Quantum-enhanced methods remain at an early stage, with potential applications in electronic structure, property prediction, and force-field computation.The review notes that classical force-field methods have efficiently modeled only small systems, while quantum methods may augment capabilities for large, strongly correlated systems.Source: §7 · p. 82
  • Open questions include interpreting model generalization as model size grows and clarifying whether critical points relate to choices of hyperparameters, cost functions, optimizers, and learning ability.The review calls for further study of models with self-organized criticality.Source: §7 · pp. 82–83
  • Open challenges include efficient data loading, noise resilience and benchmarking, finite-depth quantum-model universality, and mitigating barren plateaus.Early quantum algorithms for PCA and clustering assumed qRAM for efficient data encoding; the review also identifies exponentially vanishing gradients as an unresolved trainability concern.Source: §7 · p. 83
  • The review omits chemical-reaction modeling, where quantum computing’s role remains unexplored, and open-system dynamics despite reported ML predictions of long-time behavior.The cited examples include spin-boson dynamics and a coupled two-level system; CNNs are also reported to predict system-operator averages accurately.Source: §7 · pp. 83–84
  • Physics-inspired machine learning is another unaddressed area, although it is beginning to gain attention elsewhere and remains sparsely adopted in chemistry.Source: §7 · p. 84
Loading 2111.00851v2…