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When Quantum Meets AI: Quantum Methods for Machine Learning and Machine Learning Methods for Quantum Systems

Tak Hur

arXiv:2609.25641v1quant-phcs.AI

TL;DR

Quantum error correction still faces a critical classical bottleneck in decoding. This thesis studies learned quantum representations and methods for quantum systems, reporting greater class-ensemble separation alongside higher classification accuracy on IBM hardware.

  • Problem

    Quantum error correction still faces a critical classical bottleneck: the decoder that interprets measured syndromes.

    Source: §4.1 · p. 94

  • Method

    The thesis introduces Neural Quantum Embedding and a Mamba surface-code decoder, and frames stochastic reconfiguration using variational-state-manifold geometry.

    Source: §2.1.2 · p. 47, §4.4 · p. 114, §5.1 · p. 118

  • Results

    From approximately 0.27 to 0.84, class-ensemble distance increased, while IBM-hardware accuracy rose from 52.7% to 96.1%.

    Source: CONCLUSION · p. 133

  • Takeaways & Limitations

    In the studied hardware setting, increased class-ensemble separation coincided with higher classification accuracy.

    Source: CONCLUSION · p. 133

    Uniform bounds may be too vacuous for QML when models are expressive enough to fit random labels.

    Source: §3.2.2 · p. 79

Abstract

from arXiv · show

This thesis studies the intersection of quantum computing and artificial intelligence in two directions: quantum methods for machine learning and machine learning methods for quantum systems. For quantum machine learning, Neural Quantum Embedding learns data representations that increase the trace distance between embedded class ensembles, lowering an embedding-dependent bound on empirical risk and improving classification on noisy quantum hardware. A training objective based on the Hilbert-Schmidt inner product extends this approach to deterministic quantum computation with one qubit (DQC1) and is demonstrated on an NMR quantum processor. A margin-based generalization analysis then connects quantum neural network performance to quantum state discrimination. In the studied benchmarks, margin distributions predict generalization more reliably than parameter-count metrics. For quantum systems, a Mamba-based neural decoder for surface codes matches a reproduced Transformer baseline in memory experiments while reducing inference-cost scaling from quartic to quadratic in code distance. Under an explicit decoder-induced-noise model, it achieves lower logical error rates and a higher effective threshold. For neural quantum states, stochastic reconfiguration is interpreted as tangent-space ridge regression, with its diagonal shift controlling the bias-variance trade-off under finite Monte Carlo sampling. Multi-shift stochastic reconfiguration reduces checkpoint-local validation residuals and update variance relative to fixed-shift SR, at additional computational cost. Together, these contributions show how learned representations, statistical control, and hardware constraints shape the exchange between quantum computing and machine learning.

2.6 Expressibility and trainability diagnostics for embeddings with and without

Across embeddings, quantum decoding, and neural quantum-state optimization, the thesis applies learned representations and statistical control to problems at the intersection of quantum computing and machine learning.

2.6 Expressibility and trainability diagnostics for embeddings with and without

  • Both NQE variants are less expressive than the conventional embedding, while their larger off-diagonal kernel variance indicates entries are not exponentially concentrated.The passage interprets the higher kernel variance as improving quantum-kernel trainability.Source: §2.6 · p. 11
  • Margin-based metrics track generalization gaps more closely than effective-parameter metrics, while increased label randomization shifts margins left and widens the gap.In the reported experiments, parameter-based metrics show inconsistent or opposite trends compared with the generalization gap.Source: §2.6 · p. 12
  • Across the tested embedding schemes and datasets, higher class-ensemble trace distance corresponds to larger attainable margins and higher test accuracy, with small generalization gaps.For multi-channel NQE, trace distance also shows a positive trend with QCNN accuracy across configurations.Source: §2.6 · pp. 12, 15

Machine Learning Methods for Quantum Systems

  • Under decoder-induced noise, Mamba reaches an effective threshold of 0.0104 versus 0.0097 for Transformer and achieves lower logical error rates.The threshold comparison uses the stated latency model.Source: §2.6 · p. 13, Machine Learning Methods for Quantum… · p. 18
  • At N_s = 4096 and λ = 10^-9, matched-noise diagnostics expose ViT overfitting at comparable amplitude-gap variance, while matched-state diagnostics show lower ViT gap variance and excess risk.The plotted predictions and risks exclude the phase channel.Source: §2.6 · p. 13

2.6 Expressibility and trainability diagnostics for embeddings with and without

  • Supplementary tests report improved downstream loss across three alternative feature maps, trace-distance increases of several orders for many LIT-PCBA targets, and a positive trace-distance–accuracy trend for multi-channel NQE.The LIT-PCBA comparison covers balanced and imbalanced activator/inactivator settings.Source: §2.6 · pp. 14–15

Machine Learning Methods for Quantum Systems

  • The SR diagonal shift acts as a statistical spectral filter that trades bias against sampling variance; validation residuals are U-shaped as the shift varies.The reported multi-batch variance decreases with stronger regularization.Source: §2.6 · p. 13, Machine Learning Methods for Quantum… · p. 18
  • Multi-shift SR reduces checkpoint-local validation residuals and update variance relative to standard SR at the training shift, at additional computational cost.The reported checkpoint-local ablations use K = 4 and a shared NTK-quantile grid for multi-shift methods.Source: §2.6 · p. 14, Machine Learning Methods for Quantum… · p. 18

INTRODUCTION TO QUANTUM MACHINE LEARNING

This section introduces the foundations and scope of quantum machine learning, covering supervised quantum circuit models, structured QCNNs, quantum kernels, and QNN gradients. It frames the thesis’s study of representation limits on training loss, margins and generalization, optimization, and expressivity–generalization trade-offs, and situates these topics within the thesis’s broader directions.

Qubits and Quantum States

  • An n-qubit system occupies a 2^n-dimensional state space, and its tensor-product structure permits entangled states that cannot be factored into individual qubit states.A qubit is a two-level system that can exist in a superposition of computational basis states.Source: Qubits and Quantum States · pp. 20–21

Quantum Gates and Circuits

  • Quantum circuits compose unitary gates, and single-qubit gates together with CNOT gates can decompose any n-qubit unitary.The circuit model is the computational model used throughout the thesis.Source: Quantum Gates and Circuits · pp. 21–22

Measurement

  • Measurement extracts classical outcomes probabilistically, so quantum algorithms must be designed to produce desired answers with high probability.The thesis primarily uses computational-basis measurements.Source: Measurement · pp. 22–23

The Promise of Quantum Computation

  • An exponentially large state space alone does not guarantee computational advantage, while noisy, coherence-limited NISQ devices motivate shallow hybrid quantum-classical methods.The thesis describes current NISQ devices as having tens to hundreds of qubits, limited coherence times, and noisy gate operations.Source: The Promise of Quantum Computation · p. 23

1.1.2 Introduction to Machine Learning

  • The thesis focuses on supervised learning, in which models learn input-to-output mappings from labeled training data [5].Machine learning is framed as algorithms improving task performance through experience.Source: §1.1.2 · p. 24

From Classical Methods to Deep Learning

  • SVMs use kernels to map data into high-dimensional spaces and offer generalization guarantees grounded in statistical learning theory [6, 7].The thesis revisits kernel and margin theory for quantum generalization analysis.Source: From Classical Methods to Deep Learning · p. 24

The Transformer Era

  • Transformers [16] replaced recurrence with parallel self-attention, resolving RNN sequential bottlenecks and supporting large-scale representation learning across fields.Their use extends beyond language to computer vision, protein structure prediction, and scientific computing.Source: The Transformer Era · p. 25
  • Part II builds on Transformer-based quantum decoding and analyzes stochastic reconfiguration for neural quantum states as a statistical spectral filter for variational Monte Carlo.Chapter 4 replaces attention in the AlphaQubit architecture with a Mamba-based state-space model; Chapter 5 analyzes stochastic reconfiguration in transformer and foundation neural quantum states.Source: The Transformer Era · p. 25
  • The Transformer Era motivates asking whether quantum computing can provide model classes with inductive biases advantageous for particular learning problems.Source: The Transformer Era · p. 26

Sources of Prediction Error

  • Excess risk is organized into approximation, estimation, and optimization errors, framing analyses of embedding-induced training-loss floors and margin-based generalization.Source: Sources of Prediction Error · pp. 26–27

1.2.1 History

  • The cited work reports two proposals: in the 2010s, Rebentrost et al. [23] proposed a quantum support vector machine (QSVM), and Lloyd et al. [24] introduced quantum principal component.Source: §1.2.1 · p. 27

1.2.2 Quantum Neural Networks

  • A supervised QNN encodes classical inputs into quantum states, applies a trainable parameterized circuit, and obtains predictions by measuring an observable.Source: §1.2.2 · pp. 28–29

Structured QNNs: Quantum Convolutional Neural Networks

  • QCNNs [32] alternate local, shared-parameter convolution and pooling layers that progressively reduce qubits; their hierarchical structure avoids barren plateaus [34], though some variants are classically simulable.Prior work also applied QCNNs to classical classification by adding a data-encoding circuit [33].Source: Structured QNNs: Quantum Convolutional… · p. 30

Gradient Computation: The Parameter-Shift Rule

  • For Pauli rotation gates, the parameter-shift rule [37, 38] computes expectation-value derivatives exactly using two circuit evaluations.Source: Gradient Computation: The… · p. 31
  • The parameter-shift rule yields analytical gradients from shifted circuit evaluations, making gradient-based training and optimizers such as gradient descent and Adam compatible with quantum hardware that provides expectation values.The gradients are not finite-difference approximations.Source: Gradient Computation: The… · p. 32

The Barren Plateau Problem

  • Barren plateaus [39] cause gradient variance to vanish exponentially with qubit count, requiring exponentially many measurement shots to distinguish gradients from statistical noise; depth, global costs, noise, and entanglement can contribute.Such plateaus can negate potential quantum advantage.Source: The Barren Plateau Problem · p. 32

Quantum Feature Maps

  • Classical kernels implicitly map inputs into feature spaces and compute inner products, motivating quantum feature maps that encode inputs as states whose squared overlaps can be measured on quantum hardware.The quantum kernel is defined by the squared overlap of embedded states.Source: Classical Kernel Methods · p. 33, Quantum Feature Maps · p. 34

The ZZ Feature Map

  • Havlicek et al.’s [43] ZZ feature map encodes inputs into data-dependent entanglement and multi-qubit correlations: exact classical kernel computation is exponential in qubits, while quantum overlap estimation is polynomial.The feature map uses repeated data-encoding unitaries, and its kernel is estimated with an overlap-measurement circuit.Source: The ZZ Feature Map · pp. 34–35

Constructed Quantum Advantage Results

  • Under discrete-log intractability, quantum kernels learn a constructed distribution where polynomial-time classical learners do no better than random, even with finite-shot noise; real-world advantage remains unknown [44].The separation is established under distributional and computational assumptions; the passages do not establish that it extends to natural datasets.Source: Constructed Quantum Advantage Results · p. 35

Relationship to Quantum Neural Networks

  • With fixed data encodings, QNNs [45] are linear models in quantum feature space; kernel methods search a larger hypothesis class that may overfit, while parameterized observables constrain and regularize the model.The passages frame this as a trade-off between expressivity and generalization.Source: Relationship to Quantum Neural Networks · p. 36

Data Re-uploading and Model Expressivity

  • Data re-uploading circuits can approximate arbitrary continuous functions [46], while an equivalent kernel model for a D-layer circuit requires Ω(D) additional ancilla qubits [47].The equivalence between QNNs and kernel methods breaks down with repeated data encoding.Source: Data Re-uploading and Model Expressivity · p. 37

1.3 Summary

  • Quantum kernel methods offer rigorous supervised-learning separations under explicit assumptions, while the kernel framework clarifies when quantum neural networks behave linearly in quantum feature space.Source: §1.3 · p. 38
  • Neural Quantum Embedding trains embeddings to increase quantum-state distinguishability and address embedding-induced training-loss limits in supervised quantum learning.Source: §1.3 · p. 39
  • Margin-based bounds for quantum neural networks provide statistical control with finite data and connect trace-distance representations to quantum state discrimination.Source: §1.3 · pp. 39–40
  • Neural decoders for quantum error-correcting codes are studied through their accuracy–latency trade-off in real-time decoding benchmarks.Source: §1.3 · p. 40
  • For neural quantum states, stochastic reconfiguration is recast as tangent-space ridge regression, with finite-sample residual noise and multi-shift SR as a richer spectral filter.The chapter also analyzes the optimizer that makes neural representations of quantum many-body states practical.Source: §1.3 · p. 40

NEURAL QUANTUM EMBEDDING

This section develops Neural Quantum Embedding, using learned classical preprocessing to make embedded class ensembles more distinguishable for classification. It also examines generalization, expressibility, and trainability, and extends the method to a DQC1-measurable Hilbert–Schmidt objective with reported quantum-hardware experiments.

2.1.1 Empirical Risk and Trace Distance

  • The best achievable binary-classification error is controlled by the trace distance between embedded class ensembles, which subsequent quantum processing cannot improve.Thus, the embedding sets a floor on empirical risk before the trainable classifier acts.Source: §2.1.1 · p. 44

2.1.2 Limitations of Deterministic Embeddings

  • On balanced MNIST digits 0 vs. 1 with 4 qubits, the data-driven embedding achieved D_tr ≈ 0.84 and lowered the empirical-risk bound from L_S ≥ 0.36 to L_S ≥ 0.08 versus conventional ZZ.The same pattern held on Fashion-MNIST at 4, 8, and 12 qubits and was more pronounced under hardware noise.Source: §2.1.2 · p. 46, Figure 2.1 · p. 47

2.2.1 The NQE Framework

  • NQE [48] places a trainable classical neural network before a fixed quantum embedding to learn data-dependent preprocessing that increases class-state distinguishability and escapes the contractive bound on trainable quantum layers.The network maps inputs to embedding rotation angles and can reduce dimensionality before quantum states are prepared.Source: §2.2 · p. 47, §2.2.1 · p. 48
  • For the ZZ feature map, NQE replaces fixed angle functions with learned functions and extends to embedding circuits including amplitude and angle encoding.When the learned function can represent the original preprocessing map, the fixed embedding is recovered as a special case; improvement beyond it reflects added expressivity.Source: §2.2.1 · p. 48

2.2.2 Training via Implicit Fidelity Loss

  • NQE’s pairwise fidelity objective clusters same-class states and reduces cross-class fidelities, pushing class states toward near-pure, near-orthogonal representatives that increase trace distance and lower the empirical-risk bound.Directly optimizing trace distance would require computationally prohibitive tomography of class-averaged density matrices.Source: §2.2.2 · pp. 49–51
  • A DQC1-compatible NQE variant writes the training loss using Hilbert–Schmidt inner products between feature-map unitaries.This adapts the training objective beyond the pairwise pure-state fidelity formulation.Source: §2.2.2 · p. 51

2.2.3 Experimental Results

  • For balanced 4-qubit MNIST 0-vs-1, noiseless accuracy was 84.5±8.0% (conventional ZZ), 98.8 ± 0.0% (PCA-NQE), and 99.9 ± 0.0% (NQE).The results are summarized in Table 2.1.Source: Table 2.1 · p. 52, §2.2.3 · p. 52
  • Across noiseless and noisy MNIST and Fashion-MNIST experiments, NQE achieved lower training loss and higher classification accuracy than all tested trainable-unitary variants.The passage attributes the noisy advantage to the deeper trainable-unitary circuits’ greater noise susceptibility, while NQE’s classical preprocessing adds no quantum circuit depth.Source: §2.2.3 · p. 53, Figure 2.4 · p. 54
  • On real IBM hardware, accuracy was 52.7 ± 3.3% (conventional ZZ), 96.1 ± 0.5% (PCA-NQE), and 90.1 ± 4.5% (NQE).The passage attributes the robustness difference to larger trace distance: noise reduces separation, but PCA-NQE’s embedded states remain well-separated.Source: §2.2.3 · p. 53
  • NQE’s noisy real-hardware empirical risk fell below the noiseless theoretical lower bound for the unoptimized ZZ feature map.For this task, the text concludes that improving the embedding can matter more than reducing moderate hardware noise.Source: §2.2.3 · pp. 53–54

2.2.4 Generalization, Expressibility, and Trainability

  • The section reports numerical evidence that NQE improves generalization and trainability, assessed through effective-dimension and weight-norm measures, and expressibility and kernel-variance measures, respectively.It notes that high expressibility can be detrimental because it induces barren plateaus and kernel concentration.Source: §2.2.4 · p. 55
  • Across 200 instances, NQE reduced four-qubit QNN local effective dimension to ≈10.4–13.5 from the conventional embedding’s ≈18.5 plateau, consistent with better expected generalization.The reduction held across the tested dataset sizes and all 200 instances.Source: §2.2.4 · pp. 55–56, Figure 2.5 · p. 57
  • Across every regularization weight λ, PCA-NQE and NQE both lower weight-norm complexity G = ||W*||F/√N than the conventional ZZ feature map without NQE, yielding a tighter upper bound on generalization.The comparison reports means and one standard deviation over five independent draws of 1,000 MNIST samples.Source: Figure 2.5 · p. 57

2.3.1 The DQC1 Model

  • DQC1’s polarized probe and maximally mixed register suit NMR processors, where spatial averaging can prepare an effective DQC1 state.NQE’s Hilbert–Schmidt reformulation extends the method to ensemble platforms and was demonstrated experimentally on an NMR quantum processor.Source: §2.3 · p. 59, §2.3.1 · p. 61

2.3.2 NQE Loss Function via Hilbert–Schmidt Inner Product

  • The NQE-DQC1 objective replaces fidelity with the measured real part of the Hilbert–Schmidt inner product, driving same-class pairs toward overlap 1 and different-class pairs toward low overlap.DQC1 directly measures the required Hilbert–Schmidt inner product, while the classical neural network is optimized by standard gradient descent.Source: §2.3.2 · p. 62
  • The Hilbert–Schmidt inner-product objective makes NQE applicable to ensemble quantum systems and can be evaluated from repeated DQC1 probe-qubit measurements.The DQC1 formulation uses a single probe qubit and a mixed encoding register.Source: §2.3.2 · p. 62, §2.4 · pp. 67–68

2.3.3 Experimental Demonstration

  • In NMR training, loss converged to approximately zero by iteration 10, experiments closely matched simulations, and train- and test-set trace distances increased.The rising trace distance was consistent with the Hilbert–Schmidt surrogate aligning with improved distinguishability.Source: §2.3.3 · p. 63, Figure 2.8 · p. 64
  • NQE preprocessing raised NMR MNIST classification accuracy from 54% to 98% on the 500-sample dataset.Without NQE, the 54% accuracy was close to random guessing; Figure 2.10 presents classification outputs for the two conditions.Source: §2.3.3 · p. 65, Figure 2.10 · p. 66
  • NQE parameters learned through the NMR-compatible DQC1 protocol transferred without retraining to IBM superconducting processors, with experimental trends agreeing with simulations [49].Additional validation on Fashion-MNIST and satellite-image datasets supported applicability across the tested data domains.Source: §2.3.3 · p. 66

2.4 Summary

  • For binary quantum classification with linear loss, empirical risk is lower bounded by the trace distance between embedded class ensembles, which subsequent noisy quantum processing cannot increase.Conventional deterministic embeddings are data-agnostic and do not guarantee a large trace distance for a given task.Source: §2.4 · p. 67
  • NQE inserts a classical neural network before the quantum embedding, avoiding the quantum-state contractive constraint during preprocessing and using a fidelity loss to cluster same-class states and separate cross-class states.The preprocessing acts on classical parameters rather than quantum states.Source: §2.4 · p. 67
  • On binary MNIST, PCA-NQE raised trace distance from ≈0.27 to ≈0.84 and accuracy from 52.7% to 96.1% under real hardware noise.The reported empirical risk was also lower than the noiseless Helstrom limit associated with the conventional ZZ feature map.Source: §2.4 · p. 67

GENERALIZATION IN QUANTUM MACHINE · LEARNING

The section develops a margin-based generalization framework for quantum neural networks and connects margins to quantum state discrimination. It also reviews classical generalization theory, from finite hypothesis classes to complexity measures for infinite classes.

LEARNING

  • Inspired by Bartlett et al. [68], the framework gives multiclass quantum neural network classifiers a high-probability, function-class bound sensitive to empirical margin loss.Source: LEARNING · p. 70
  • In the studied QPR setting, margin-based metrics predict generalization performance more reliably than parameter-based metrics.Source: LEARNING · p. 70
  • When other bound parameters are comparable, embeddings with larger trace distances can support larger margins and tighter margin-based generalization bounds.The section links this relationship to quantum state discrimination and notes Neural Quantum Embedding (NQE) as an example.Source: LEARNING · p. 70

3.1 Theoretical Background

  • The theoretical background sets up supervised learning with a hypothesis class, a training sample, and an unknown distribution, then progresses toward margin theory.Source: §3.1 · p. 70

3.1.1 Generalization Bounds for Finite Hypothesis Classes

  • For finite hypothesis classes, data-dependent selection motivates a union bound, whose generalization penalty grows logarithmically with the number of candidates.Source: §3.1.1 · pp. 71–72

3.1.2 Complexity Measures for Infinite Hypothesis Classes

  • Infinite hypothesis classes, including those defined by neural networks, support vector machines, and quantum circuits, require complexity measures that capture their effective capacity to fit patterns.Source: §3.1.1 · p. 72, §3.1.2 · p. 72

VC Dimension

This section develops margin-based generalization analysis for quantum neural networks, situating it alongside class-wide complexity and hypothesis-dependent bounds and comparing margins with parameter-based measures. It also links margins to quantum state discrimination, with trace distance bounding the achievable margin mean, and finds that margins track generalization more reliably in the tested settings.

VC Dimension

  • The VC dimension [7, 21] characterizes PAC learnability: a binary class is learnable exactly when its VC dimension is finite, but neural-network VC dimension scales with parameter count.This can make bounds vacuous for overparameterized networks that nevertheless generalize well in practice.Source: VC Dimension · p. 73

Rademacher Complexity

  • Rademacher complexity [69] measures how well a function class correlates with random noise, and its data-dependent generalization bound tightens with larger samples or lower complexity.Source: Rademacher Complexity · pp. 73–74

Covering Numbers and Dudley’s Entropy Integral

  • Covering numbers approximate even infinite function classes with finite sets at a chosen precision, and Dudley’s entropy integral [21] connects these numbers across scales to Rademacher complexity.The chapter uses this integral to derive its margin generalization bound.Source: Covering Numbers and Dudley’s Entropy… · pp. 74–75

3.1.3 Margin Theory

  • Margin theory complements class-wide bounds with a hypothesis-dependent view: larger classification margins indicate more confident predictions, and classical bounds use Lipschitz surrogate loss in place of 0-1 loss.Source: §3.1.3 · p. 75, Classical Margin Bounds · pp. 75–76

Margin Bounds for Deep Networks

  • Bartlett et al. [68] showed that spectrally normalized margin distributions correlate with deep-network generalization, offering an alternative to parameter-count or VC-dimension bounds.Source: Margin Bounds for Deep Networks · p. 76
  • The chapter extends margin-based generalization analysis from classical neural networks to quantum neural networks, adapting covering-number techniques.Source: Margin Bounds for Deep Networks · p. 77

3.2.1 Uniform Generalization Bounds

  • Reviewed quantum uniform bounds use trainable-parameter counts [74], Fisher-information effective dimension [57], quantum Rademacher complexity [75, 76, 77], and information-theoretic quantities [78, 79].Source: §3.2.1 · pp. 77–78

3.2.2 Limitations of Uniform Bounds

  • Quantum neural networks, including QCNNs, can fit randomized labels on quantum datasets despite small qubit and parameter counts, exposing the limits of uniform bounds.Source: §3.2.2 · p. 79

3.3.2 Margin Generalization Bound

  • The QNN margin-loss Rademacher complexity is bounded using channel components, Lipschitz continuity, unitary-family covering numbers, and Dudley’s entropy integral.The derivation adapts matrix covering techniques to quantum circuits.Source: §3.3.2 · p. 81
  • The bound includes empirical margin loss and a complexity term scaling as 1/γ, so larger training margins can yield a tighter bound.The bound is stated for an n-qubit QNN whose unitaries lie within radius b of a reference unitary, with probability at least 1 − δ over an i.i.d. sample.Source: §3.3.2 · p. 82

3.2.2 Limitations of Uniform Bounds

  • The bound also depends on POVM spectral norms, making generalization sensitive to how information is extracted from the quantum state.The section motivates this hypothesis-dependent approach by noting that margin metrics were strong predictors in classical deep learning [71, 72, 73].Source: §3.2.2 · p. 79, §3.3.2 · p. 83

Margin Distribution and Generalization Performance

  • Larger, right-skewed margins aligned with higher test accuracy and tighter bounds, while label randomization shifted margins left and increased generalization gaps and bounds.The figure presents optimized 8-qubit QCNNs with one, five, or nine layers under 0%, 50%, and 100% label randomization.Source: Figure 3.1 · p. 85, Margin Distribution and Generalization… · p. 85

Quantum Phase Recognition

Predicting the Generalization Gap: Margins vs. Parameters

  • Margin metrics tracked the generalization gap more closely than effective-parameter metrics across the studied QCNN experiments.Figure 3.2 compares generalization gap, inverse median margin, and effective parameters with threshold 10^-2 across layer count, label randomization, and variational ansatz.Source: Figure 3.2 · p. 86, Margin Distribution and Generalization… · p. 86
  • Inverse median margin tracks the generalization-gap peak at five QCNN layers, whereas effective parameters rise monotonically and miss that peak.The margin also captures variations across the tested hyperparameters and variational ansätze.Source: Predicting the Generalization Gap:… · p. 87
  • Margin-based metrics correlate more strongly with the generalization gap than parameter-based metrics by both mutual information and Kendall rank correlation.Figure 3.3 compares margin-distribution metrics with total and effective parameter counts.Source: Predicting the Generalization Gap:… · p. 87, Figure 3.3 · p. 88, §3.4 · p. 91

3.3.4 Connection to Quantum State Discrimination

  • Trace distance upper-bounds the achievable margin mean, so embeddings that increase class-ensemble distinguishability, including NQE, permit larger margins and connect state distinguishability to generalization [48, 89, 90].The text identifies the Helstrom measurement as the equality case for the bound.Source: §3.3.4 · p. 89
  • Across fixed, trainable, and NQE embeddings, increasing trace distance shifts margins rightward; larger, right-skewed margins accompany higher test accuracy while generalization gaps stay small.These experiments use optimized 8-qubit QCNNs on MNIST, Fashion-MNIST, and Kuzushiji-MNIST.Source: §3.3.4 · pp. 89–90, Figure 3.4 · p. 90

3.4 Summary

  • The multiclass QNN generalization bound depends on the margin distribution and yields tighter estimates than parameter-count bounds when training margins are large.In the empirical memorization comparisons studied, parameter-count bounds are often too vacuous.Source: §3.4 · p. 91

NEURAL DECODERS FOR QUANTUM ERROR · CORRECTION

These parts examine Mamba-based neural decoding for surface-code error correction, focusing on the balance between logical-error performance and inference cost. They describe replacing attention with Mamba, reporting comparable accuracy on memory data, improved inference scaling, and a higher finite-size effective threshold under specified latency or decoder-induced-noise models.

CORRECTION

  • Mamba [96] uses an O(d^2) selective scan in code distance, preserving comparable decoding accuracy to the reproduced Transformer baseline while improving on O(d^4) attention scaling; under the chapter’s simulated latency model, it yields a higher finite-size effective threshold.Source: CORRECTION · pp. 93–94

4.1 Introduction to Quantum Error Correction

  • Quantum error correction protects encoded logical qubits, but decoding noisy syndrome measurements fast enough to keep pace with their production remains a scaling bottleneck [95].The decoder must maintain throughput comparable to syndrome generation so the classical backlog remains bounded.Source: §4.1 · pp. 94–95

4.1.1 Surface Codes

  • The chapter focuses on rotated surface codes, a leading near-term fault-tolerance candidate using nearest-neighbor interactions and local measurements; a distance-d patch encodes one logical qubit in 2d^2 − 1 physical qubits.The patch comprises d^2 data qubits and d^2 − 1 measurement qubits.Source: §4.1.1 · p. 95

4.1.2 The Decoding Problem

  • Decoding infers a correction from syndrome history by identifying an error equivalence class; because maximum-likelihood decoding is #P-hard in general [107], efficient approximate methods are needed.Real-time processors need decoding throughput comparable to syndrome generation to keep backlog bounded [95].Source: §4.1.2 · pp. 96–98

4.2.1 Overview of Neural Decoder Literature

  • Because delayed corrections can add effective noise, decoding speed is an essential practical companion to accuracy in quantum error correction.The section identifies an ongoing trade-off: highly accurate neural decoders can require large models and expensive inference, while lightweight decoders may sacrifice accuracy.Source: §4.1.2 · p. 98, §4.2.1 · p. 101

4.1.2 The Decoding Problem

  • Data-driven decoding pretrains neural decoders on synthetic data, then fine-tunes them on hardware data to learn device-specific noise without an explicit analytical noise model.The cited noise characteristics include cross-talk, leakage, and measurement-induced errors.Source: §4.1.2 · pp. 98–99

4.2.2 AlphaQubit

  • 2.901 × 10^-2 at distance 3 and 2.748 × 10^-2 at distance 5 were AlphaQubit’s logical error rates, with error suppression ratio Λ = 1.056.The section describes these Sycamore results as competitive with the strongest tested decoders.Source: §4.2.2 · p. 102
  • At distance 9, AlphaQubit latency is approximately 40 μs, versus a ∼1 μs superconducting-qubit QEC cycle; attention scales quadratically in stabilizer count and quartically in code distance.The stabilizer count scales as d^2, making attention’s per-cycle cost O(d^4).Source: §4.2.2 · p. 103

4.3 Mamba Decoder

  • The proposed Mamba decoder replaces AlphaQubit’s attention mechanism with a selective state-space module of linear complexity while preserving comparable accuracy.State-space models are presented as sequence-model alternatives to Transformers that offer linear-time complexity.Source: §4.2.2 · p. 103, §4.3 · p. 103, §4.3.1 · p. 103

4.3.1 Mamba and State Space Models

  • Mamba [96] makes B, C, and Δ input-dependent for content-aware filtering, and its associative recurrence enables parallel-prefix evaluation in O(log L) parallel depth.The input-dependent parameters let the model decide which information to propagate through the hidden state or discard.Source: §4.3.1 · p. 105

4.3.2 Architecture and Training

  • The decoder replaces attention with a Mamba module in each Syndrome Mixer; Sycamore training uses pretraining then fine-tuning, while simulated real-time decoding trains from scratch.The decoder retains a Stabilizer Embedder, RNN Core, and Readout Network.Source: §4.3.2 · pp. 106–108

4.3.3 Experimental Results

  • The evaluation covers Sycamore memory experiments on real hardware data and simulated real-time decoding with decoder-induced noise.Source: §4.3.3 · p. 109

Memory Experiments on Sycamore Data

  • On the Sycamore memory experiment, Mamba’s LER is approximately 2.98 × 10−2 at distance 3 and 3.03 × 10−2 at distance 5, tracking the reproduced Transformer, beating matching-based decoders but trailing the tensor-network decoder (ε = 2.915 × 10−2).The models were pretrained on 100 million samples due to computational constraints.Source: Memory Experiments on Sycamore Data · pp. 109–110, Figure 4.3 · p. 110

Inference Speed

  • ≈3×: the Mamba block reaches roughly three times the speed of the MHA block at d=39, with curves deviating near d≈21 and the advantage widening as distance grows.Inference time was measured on an NVIDIA RTX 4090 GPU.Source: Inference Speed · pp. 110–111, Figure 4.3 · p. 110

Real-Time Decoding with Decoder-Induced Noise

4.4 Summary

  • On Google Sycamore memory experiments, Mamba’s logical error rates were ε≈2.98 × 10−2 (distance 3) and ε≈3.03 × 10−2 (distance 5), closely tracking the reproduced Transformer.Source: §4.4 · p. 114

Inference Speed

  • Inference followed O(d^2) for Mamba versus O(d^4) for Transformer, reaching roughly 3× faster at d=39; constant-factor overhead masks the advantage at d≤5.The scaling gap yields larger latency reductions at larger code distances.Source: Inference Speed · p. 111, §4.4 · p. 114

NEURAL QUANTUM STATES

These studies examine stochastic reconfiguration for neural quantum states as a statistical method whose diagonal shift balances fit against variance under finite Monte Carlo sampling. They also assess multi-shift SR using held-out or checkpoint-local comparisons, including its limitations and added update-construction cost.

NEURAL QUANTUM STATES

  • Multi-shift SR combines updates with different shifts to make better use of the available Monte Carlo samples.The supplied passages introduce this variant but do not report its empirical results in this part.Source: NEURAL QUANTUM STATES · p. 116

5.1 Neural Quantum States and Stochastic Reconfiguration

  • Modern NQS can have parameter counts one to three orders of magnitude above sample counts, so minSR solves in sample space but still incurs O(N_s^2P) kernel formation and O(N_s^3) solving costs.SR also corrects the mismatch between Euclidean parameter movement and changes in the represented state by using variational-state-manifold geometry.Source: §5.1 · pp. 118–119

5.2 SR as Statistical Spectral Filtering

  • The diagonal shift controls a bias–variance trade-off: larger shifts suppress noisy small-eigenvalue directions but increase bias by shrinking useful update components.It is therefore a statistical regularizer, not merely a numerical tolerance.Source: §5.2 · p. 122
  • The filtering view predicts a U-shaped validation residual as the shift varies, while batch-to-batch update variance decreases monotonically as finite-batch noise is suppressed.Held-out tangent-feature predictions provide these diagnostics at large scale; validation residual tracks excess risk plus the expressivity-gap term.Source: §5.2 · pp. 122–123

5.3 Evidence Across System Scales

  • Overparameterization can help when it reduces the expressivity gap, but hurt when finite-sample SR overfits a comparable gap.In matched-noise tests, RBM and ViT excess risks were approximately 0.53 ± 0.08 and 3.39 ± 0.73; in matched-state tests, ViT reduced gap variance from 1.95 to 0.816 and excess risk from approximately 2.53 ± 0.19 to 1.59 ± 0.08.Source: §5.3 · p. 124
  • On the L=100 TFIM foundation NQS, validation residual is U-shaped in the SR shift, while multi-batch variance decreases as the shift increases.All shifts were evaluated at the same fixed checkpoints, isolating the ridge filter’s statistical effect rather than differences between training trajectories.Source: Figure 5.2 · p. 126

5.4 Multi-Shift Stochastic Reconfiguration

  • MS-SR mixes shifted solves on independent batches, giving a richer spectral filter than single-shift SR and flexibility to suppress noisy low-eigenvalue directions.Its solve cost is O(KN_s^3), versus O(K^3N_s^3) for one solve on a batch of size KN_s; it is most worthwhile when finite-batch variance is a visible bottleneck.Source: §5.4 · pp. 126–127
  • MS-SR reduces validation residual and multi-batch variance by 39% and 43% on TFIM, and 14% and 66% on J1–J2, versus standard SR.Oracle-tuned bagging reduces those metrics by 45% and 49% on TFIM, and 18% and 80% on J1–J2, yielding lower mean values than MS-SR in both systems.Source: §5.4 · p. 128, Figure 5.3 · p. 129
  • On TFIM, MS-SR has 27% lower residual than training-shift bagging, but a well-chosen single shift can beat MS-SR, so it is not uniformly superior to tuned bagging.On J1–J2, mean residuals are 3.98 for training-shift bagging, 3.92 for oracle-tuned bagging, and 4.13 for MS-SR; MS-SR adapts across regularization scales without a separate validation-based shift search.Source: §5.4 · p. 129
  • MS-SR achieved lower energy in 3 of 5 paired continuations, but the mean paired difference was −1.78×10−5 with a 95% interval including zero.The comparison conditions on continuations from one shared SR checkpoint, so it does not establish a reliable final-energy advantage across independently trained source states or at equal wall time.Source: §5.4 · p. 130, Figure 5.4 · p. 131
  • MS-SR takes about four times as long as one SR update and adds approximately 1.03% over bagged SR in the reported update-construction benchmark.The benchmark measures steady-state update-construction cost on one H100 80GB GPU; the table reports medians over three paired device/seed settings.Source: §5.4 · p. 131, Table 5.1 · p. 132

5.5 Summary

  • At a fixed wave function, SR is ridge regression from tangent features to centered local energy, and its diagonal shift controls a bias–variance trade-off under finite sampling.Finite batches make the population-orthogonal tangent-space expressivity gap behave like residual noise that empirical SR can fit; increasing the shift suppresses variance while shrinking useful components.Source: §5.5 · p. 132
  • Foundation-NQS diagnostics show a U-shaped validation residual and decreasing multi-batch variance as the solve-time shift increases.Exact 4 × 4 amplitude-regression experiments found that a larger tangent space helps when it reduces the expressivity gap, but can hurt when it overfits a comparable gap.Source: §5.5 · p. 132
  • Independent multi-shift solves reduce validation residuals and update variance relative to fixed-shift SR, at additional computational cost.This result motivates MS-SR as an application of the spectral-filtering view of SR.Source: §5.5 · p. 132

CONCLUSION

The thesis presents a two-way exchange between quantum computing and AI: learned representations and statistical control support quantum tasks, while quantum settings challenge AI with new mathematical objects and physical constraints. Across its chapters, it addresses representation, generalization, decoding latency, and finite-sample optimization.

CONCLUSION

  • On binary MNIST, Neural Quantum Embedding raised class-ensemble trace distance from approximately 0.27 to 0.84 and IBM-hardware accuracy from 52.7% to 96.1%.Its DQC1/NMR extension reached 98% accuracy, compared with 54% without NQE.Source: CONCLUSION · p. 133
  • Margin-based metrics predicted generalization in quantum phase recognition more reliably than parameter-based metrics, and margin mean connects to trace distance.The connection implies that embeddings with greater state distinguishability can make larger margins attainable.Source: CONCLUSION · p. 133
  • On Sycamore memory-experiment data, Mamba matched the reproduced Transformer baseline’s accuracy while scaling inference cost as O(d^2) rather than O(d^4).With simulated real-time decoding and decoder-induced noise, the latency advantage became an accuracy advantage.Source: CONCLUSION · pp. 133–134
  • Multi-shift stochastic reconfiguration reduced validation residual and multi-batch variance versus standard SR by approximately 39%/43% on TFIM and 14%/66% on J1–J2.At a fixed wave function, SR is tangent-space ridge regression, and its diagonal shift acts as a statistical spectral filter rather than only a numerical stabilizer.Source: CONCLUSION · p. 134
  • In five paired online continuations from one shared 8×8 J1–J2 checkpoint, independent endpoint evaluations favored MS-SR in three pairs, with mean energy difference per site −1.78 × 10−5.Source: CONCLUSION · p. 134
  • The thesis concludes that quantum computing and AI benefit from a two-way exchange, with AI contributing data-driven representations, scalable architectures, and statistical regularization.Source: CONCLUSION · p. 134

Appendices … A.1 Additional Details for Neural Quantum Embedding

The appendix collects selected technical details that support reading the thesis, emphasizing derivations and reproducibility for material too detailed for the main narrative.

CONTRIBUTIONS

  • The appendix collects selected technical details, emphasizing derivations and reproducibility.Source: CONTRIBUTIONS · p. 154
  • These notes support reading the thesis but are too detailed for the main narrative.Source: CONTRIBUTIONS · p. 154

A.1.1 Implicit Fidelity Loss and Trace Distance · A.2 Additional Details for Margin Generalization

The appendix explains how NQE’s pairwise fidelity objective aligns with increasing class trace distance, and reports that its classification benefit persists across alternative feature-map ansatzes. It also relates the trace-distance analysis to MSE loss and describes a DQC1-compatible multi-class extension.

A.1.1 Implicit Fidelity Loss and Trace Distance

  • The pairwise fidelity objective is a tractable proxy for increasing trace distance between class-averaged embedded states.Same-label terms push within-class fidelities toward one, increasing class purity; different-label terms push cross-class fidelities toward zero, increasing the trace-distance lower-bound side.Source: §A.1.1 · pp. 154–155
  • Improving the embedding-induced lower bound for linear misclassification loss also improves the range in which empirical MSE loss can lie.The appendix relates the trace-distance-oriented linear-loss analysis to commonly used mean-squared-error objectives.Source: §A.1.1 · p. 155
  • The NQE benefit is not specific to the ZZ feature map: all six dataset–ansatz combinations show decreasing NQE loss and increasing class trace distance.The tested alternatives use Hamiltonian-inspired feature maps at depth M=4, with MNIST and Fashion-MNIST.Source: §A.1.1 · pp. 157–158
  • On MNIST, NQE accuracies were 0.95, 0.92, and 0.99 for H_XY, H_YZ, and H_XYZ, versus without-NQE baselines of 0.49, 0.55, and 0.81.The figure shows decreasing NQE loss, increasing class trace distance, and lower downstream PQC loss than without NQE across the tested ansatzes.Source: §A.1.1 · p. 158, Figure A.1 · p. 159
  • On Fashion-MNIST, NQE accuracies were 0.87, 0.80, and 0.90 for H_XY, H_YZ, and H_XYZ, versus without-NQE baselines of 0.49, 0.54, and 0.50.These results, together with the MNIST results, show the benefit across three alternative Hamiltonian-inspired feature maps at depth M=4.Source: §A.1.1 · p. 158, Figure A.1 · p. 159
  • The NQE-DQC1 objective extends to multiple classes through one-vs-one, one-vs-rest, or a direct Kronecker-delta target while remaining DQC1-compatible.Each term requires estimating a Hilbert–Schmidt inner product between two data-dependent unitaries.Source: §A.1.1 · p. 158

A.2.1 Proof Structure and Mixed-State Extension

Theorem 3.1 combines a ramp-loss margin argument with covering-number bounds for the QNN-induced function class. The proof extends to mixed input states by vectorization, using a distance bound on U⊗U* and Frobenius norms of POVM elements.

A.2.1 Proof Structure and Mixed-State Extension

  • The proof controls the restricted ramp-loss class through covering numbers, then applies Dudley’s entropy integral to obtain the Rademacher term.The composed ramp-margin map has Lipschitz constant proportional to E/γ, and the unitary distance is measured relative to a reference unitary.Source: §A.2.1 · pp. 159–160
  • With probability at least 1−δ, expected ramp loss is bounded using empirical ramp loss, the sample Rademacher complexity of Fγ, and √ln(2/δ)/(2m).The ramp loss upper-bounds zero-one loss and is upper-bounded by empirical margin error at threshold γ.Source: §A.2.1 · p. 160
  • For mixed input states, vectorization applies the distance bound to U⊗U*, with the measurement-size term evaluated using POVM-element Frobenius norms.Source: §A.2.1 · pp. 160–161

A.2.2 Experimental Protocol Notes · A.3 Additional Details for Stochastic Reconfiguration as Spectral Filtering

The appendix records selected reproducibility details for margin-generalization experiments and experimental choices from Hur and Park [65] to help interpret Chapter 3’s margin plots.

A.2.2 Experimental Protocol Notes

  • Experimental choices from Hur and Park [65] are provided to help interpret Chapter 3’s margin plots.Source: §A.2.2 · p. 161
  • Table A.2 records selected reproducibility details for the margin-generalization experiments.Source: Table A.2 · p. 161

A.3.1 Fixed-Checkpoint Regression Identities

Fixed-checkpoint SR identities interpret regression residuals and risk through tangent-space geometry, while clarifying how complex conventions, amplitude-only diagnostics, and sample centering qualify that interpretation.

A.3.1 Fixed-Checkpoint Regression Identities

  • For complex wave functions with real parameter increments, SR is equivalent to a real least-squares system using the real parts of the QGT and force.The complex extension uses modulus squares and the corresponding real or Hermitian QGT seminorm.Source: §A.3.1 · p. 162
  • Held-out tangent residuals estimate excess risk plus a constant irreducible gap, providing the validation identity for fixed-checkpoint SR.The residual gap is defined by the population least-squares SR direction and the normal equations.Source: §A.3.1 · p. 163
  • On the non-null QGT subspace, ridge shrinkage bias and finite-sample variance yield the spectral-risk proxy.The variance scaling is derived under a scalar residual-noise approximation.Source: §A.3.1 · pp. 163–164
  • With the full QGT, regression excess risk measures discrepancy from the population tangent-space imaginary-time step in local state geometry.The interpretation is stated for the full QGT and its associated S-seminorm.Source: §A.3.1 · p. 164
  • Chapter 5's amplitude-only regression risks omit the phase channel and therefore do not measure full complex-SR error or complete local infidelity.The diagnostics use amplitude features and a real local-energy target while evaluating local energies and Born probabilities from the complex state.Source: §A.3.1 · p. 164
  • Sample centering is equivalent to ridge regression with an unpenalized intercept, removing the constant energy/normalization component but not the expressivity gap driving finite-sample overfitting.Optimizing the intercept subtracts the sample means of the log-derivatives and local energies.Source: §A.3.1 · p. 164

APPENDIX B RELATED NON-FIRST-AUTHOR CONTRIBUTIONS

The appendix summarizes two collaborative papers that extend the Neural Quantum Embedding perspective to ligand-based virtual screening and multi-channel image data.

Appendix B Related Non-First-Author Contributions

  • Two related collaborative papers extend the Neural Quantum Embedding perspective developed in Chapter 2.Table B.1 provides a compact overview of these non-first-author contributions.Source: Appendix B · p. 166, Table B.1 · p. 166
  • One paper applies trainable quantum embeddings to ligand-based virtual screening.Source: Appendix B · p. 166
  • The other generalizes Neural Quantum Embedding to multi-channel image data.Source: Appendix B · p. 166

B.1 Quantum Embeddings for Ligand-Based Virtual Screening

Choi et al. [155] evaluate optimized quantum embeddings for ligand-based screening, reporting stronger class separation on LIT-PCBA and higher balanced accuracy from quantum kernels on COVID-19 data.

B.1 Quantum Embeddings for Ligand-Based Virtual Screening

  • For balanced GBA test data, trace distance increased from approximately 5×10^-4 to about 0.55 after ZZ-map NQE training.Source: §B.1 · p. 167
  • On LIT-PCBA, ZZ-map NQE raised class-ensemble trace distances from 10^-4–10^-3 before training to 10^-1 or higher for several targets across training and test sets.Source: §B.1 · p. 167, Figure B.1 · p. 168
  • Quantum-pretrained classical embeddings often improved performance in limited-data or imbalanced cases, showing that learned representations can transfer beyond fully quantum classifiers.Source: §B.1 · p. 168
  • Balanced accuracy reached 0.83 and 0.80 with 4- and 8-qubit ZZ projected quantum kernels, versus 0.59–0.65 for classical SVMs.Source: §B.1 · pp. 168–169

B.2 Multi-Channel Convolutional Neural Quantum Embedding

CNQE extends neural quantum embedding to multi-channel images through classical convolutional interfaces that map channels into quantum feature-map parameters [156]. The interface shapes parameter efficiency and the separability of resulting quantum states.

B.2 Multi-Channel Convolutional Neural Quantum Embedding

  • CNQE introduced three classical-to-quantum interface structures, g_a, g_b, and g_c, differing in whether multi-channel inputs are processed jointly or separately before quantum encoding [156].Source: §B.2 · p. 169
  • The study compared fidelity- and Hilbert–Schmidt-based losses and multiple embedding circuits using QCNN classifiers on binary tasks from CIFAR-10 and Tiny ImageNet.Source: §B.2 · p. 169
  • Across tested configurations, class trace distance positively tracked QCNN accuracy, supporting it as a diagnostic of embedding quality and achievable classification performance.Source: §B.2 · pp. 169–170, Figure B.2 · p. 170
  • CNQE-QCNN reached near 95% accuracy on CIFAR-10 frog–ship and Tiny ImageNet school-bus–maypole in favorable configurations; channel-separated interfaces used the fewest parameters while remaining competitive.Source: §B.2 · p. 170

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