Technical Summary: When Quantum Meets AI

This dissertation investigates the bidirectional intersection of quantum computing and artificial intelligence, structured into two complementary parts: "Quantum for AI," which explores how quantum models enhance machine learning, and "AI for Quantum," which examines how machine learning techniques solve fundamental problems in quantum error correction and many-body physics.

Part I: Quantum for AI

Chapter 2: Neural Quantum Embedding (NQE)

Problem: In quantum supervised learning, the performance of a classifier is fundamentally constrained by the data embedding. The empirical risk for binary classification is lower-bounded by the trace distance between the class-averaged density matrices of the embedded data. Conventional deterministic embeddings (e.g., amplitude, angle, or fixed ZZ feature maps) are data-agnostic and often fail to produce a large trace distance for specific datasets. Furthermore, the contractive property of trace distance under quantum channels implies that subsequent noisy quantum processing cannot increase this distinguishability; poor initial embedding cannot be recovered by the classifier.

Methodology: The author proposes Neural Quantum Embedding (NQE), a framework that inserts a classical neural network $g(x, w)$ between the raw data and the quantum embedding circuit. This network learns a data-dependent preprocessing map that transforms input features into rotation angles for the quantum circuit, effectively learning an embedding that maximizes the trace distance between class ensembles.

• Training Objective: Since computing trace distance requires full tomography, NQE is trained using an implicit fidelity loss. This loss minimizes the overlap between cross-class pairs and maximizes the overlap within same-class pairs, acting as a tractable surrogate for maximizing trace distance.
• DQC1 Extension: To extend NQE to ensemble quantum systems (e.g., NMR) where pure state preparation is impractical, the fidelity-based loss is reformulated using the Hilbert–Schmidt inner product. This quantity is directly measurable via the Deterministic Quantum Computation with One Qubit (DQC1) model.

Key Results:

• Trace Distance Improvement: On the MNIST binary classification task (digits 0 vs. 1), NQE increased the trace distance from $\approx 0.27$ (conventional ZZ map) to $\approx 0.84$.
• Accuracy Gains: This improvement translated to a classification accuracy increase from 52.7% to 96.1% on noisy IBM quantum hardware, and 98% on an NMR quantum processor. Notably, the noisy empirical risk achieved with NQE fell below the noiseless theoretical lower bound of the unoptimized conventional embedding.
• Generalization and Trainability: NQE was shown to reduce the local effective dimension of the resulting quantum neural networks (improving generalization bounds) and reduce the expressibility of the embedding, thereby mitigating barren plateaus and kernel concentration issues.

Significance: The work establishes that the data embedding sets a "representation-induced floor" on training loss. By moving the optimization of this representation to the classical domain, NQE bypasses quantum contractive constraints, significantly improving performance on near-term hardware.

Chapter 3: Generalization in Quantum Machine Learning

Problem: While low training error is necessary, it does not guarantee good performance on unseen data. Existing generalization bounds for quantum models often rely on uniform measures (e.g., parameter counts, VC dimension) which are frequently too vacuous to distinguish between models that learn patterns and those that merely memorize data, especially in overparameterized regimes.

Methodology: The author develops a margin-based generalization framework for Quantum Neural Networks (QNNs), adapting classical margin theory to the quantum setting.

• Theoretical Bound: A high-probability generalization bound is derived for multiclass QNNs using covering number techniques and Dudley's entropy integral. The bound depends on the empirical margin loss and the spectral norms of the POVMs, rather than just the total parameter count.
• Connection to State Discrimination: The thesis establishes a theoretical link between the margin mean and the trace distance between class ensembles. Specifically, the achievable margin is upper-bounded by the trace distance, connecting the generalization analysis of this chapter to the embedding theory of Chapter 2.

Key Results:

• Predictive Power: Experiments on the Quantum Phase Recognition (QPR) task demonstrate that margin-based metrics (median, mean, and lower quartile of the margin distribution) are significantly stronger predictors of generalization gaps than parameter-based metrics (total parameters or effective parameters). This holds even when models are trained on randomized labels.
• Embedding Connection: Empirical validation shows that embeddings with larger trace distances (such as those produced by NQE) support larger classification margins, leading to tighter generalization bounds and higher test accuracies.

Significance: This work provides a more nuanced theoretical tool for analyzing QML generalization, moving beyond vacuous uniform bounds. It highlights that the quality of the quantum embedding (state distinguishability) directly influences the attainable margins and, consequently, the generalization capability of the model.

Part II: AI for Quantum

Chapter 4: Neural Decoders for Quantum Error Correction

Problem: As quantum processors scale, the classical decoder for quantum error correction (QEC) must process syndrome data in real-time with low latency. Traditional decoders struggle with the computational complexity of large code distances. While Transformer-based neural decoders (e.g., AlphaQubit) offer high accuracy, their self-attention mechanisms scale as $\mathcal{O}(d^4)$ with code distance $d$, creating a latency bottleneck that introduces decoder-induced noise.

Methodology: The author proposes a neural decoder based on the Mamba architecture, a selective state-space model (SSM).

• Architecture: The Mamba decoder replaces the multi-head attention mechanism of the Transformer with a selective scan operation that scales as $\mathcal{O}(d^2)$. It retains the overall recurrent structure of AlphaQubit (Stabilizer Embedder, RNN Core, Readout Network) but utilizes Mamba blocks for the Syndrome Mixer layers.
• Training: The model is trained using a two-stage pipeline: pretraining on synthetic data generated from a detector error model calibrated on Google's Sycamore processor, followed by fine-tuning on real hardware data.

Key Results:

• Accuracy: On Sycamore memory experiment data, the Mamba decoder achieved logical error rates comparable to the reproduced Transformer baseline ($\approx 2.98 \times 10^{-2}$ at distance 3).
• Latency and Threshold: The Mamba block demonstrated $\mathcal{O}(d^2)$ inference scaling compared to the Transformer's $\mathcal{O}(d^4)$. Under a simulated real-time decoding model where decoder-induced noise scales with computational complexity, the Mamba decoder achieved a higher finite-size effective threshold ($p_{th} \approx 0.0104$) compared to the Transformer ($p_{th} \approx 0.0097$).

Significance: The work demonstrates that replacing attention mechanisms with state-space models can preserve decoding accuracy while significantly reducing latency scaling. This improvement in latency translates directly to a higher tolerance for physical error rates in real-time QEC scenarios.

Chapter 5: Neural Quantum States (NQS) and Stochastic Reconfiguration

Problem: Neural Quantum States use expressive neural networks to represent many-body wave functions, optimized via Variational Monte Carlo (VMC). The standard optimizer, Stochastic Reconfiguration (SR), solves a linear system involving the Quantum Geometric Tensor (QGT). In modern, overparameterized NQS, the number of parameters often exceeds the number of Monte Carlo samples, making the QGT rank-deficient and ill-conditioned. The diagonal shift ($\lambda$) used to regularize SR is typically viewed as a numerical stabilizer, but its statistical role in this regime is not fully understood.

Methodology: The author reframes SR as statistical spectral filtering (ridge regression) in the tangent space of the wave function.

• Bias-Variance Tradeoff: The diagonal shift is shown to act as a filter that trades bias against sampling variance. In the overparameterized regime, finite-sample noise (arising from the expressivity gap) can be overfit by the optimizer. The shift suppresses this noise but shrinks useful update directions.
• Multi-Shift SR (MS-SR): To better utilize available samples, the author proposes MS-SR, which combines independent SR updates computed with different shifts. This mixture allows for a more flexible spectral filter than a single shift.

Key Results:

• Spectral Filtering Evidence: Experiments on small-scale Heisenberg models and large-scale Transverse-Field Ising models confirm that the validation residual exhibits a U-shaped dependence on the shift parameter, and multi-batch variance decreases with stronger regularization, consistent with the noisy-ridge interpretation.
• Performance: MS-SR reduced validation residuals and multi-batch variance by approximately 39%/43% (TFIM) and 14%/66% ($J_1-J_2$) relative to standard SR at the training shift. In online training continuations, MS-SR showed a tendency toward lower energies, though the advantage was not statistically significant across all independent trials.

Significance: This work provides a statistical interpretation of the SR diagonal shift, moving beyond its numerical justification. It demonstrates that in the overparameterized regime, SR acts as a spectral filter for finite-sample noise, motivating the development of multi-shift strategies to improve optimization stability and efficiency.

Overall Conclusion

The dissertation posits that the intersection of quantum computing and AI is a bidirectional exchange. Quantum computing challenges AI to operate under new physical constraints (e.g., trace distance limits, barren plateaus, latency-induced noise), while AI provides principled tools (e.g., data-driven embeddings, margin-based analysis, scalable architectures, statistical regularization) to address the hardest problems in quantum systems. The contributions collectively advance the state of the art in both quantum machine learning performance and the practical feasibility of quantum error correction and many-body simulation.