Benchmarking loss functions for trainable quantum feature maps

Technical Summary: Benchmarking Loss Functions for Trainable Quantum Feature Maps

Problem Statement
Quantum Support Vector Machines (QSVMs) rely on Quantum Feature Maps (QFMs) to encode classical data into high-dimensional quantum states. While standard QSVMs utilize fixed QFMs, this limits their expressivity and adaptability to complex, non-linear data distributions. Trainable Quantum Feature Maps (TQFMs) address this by introducing parameterized gates and data re-uploading to optimize the embedding space. However, the optimization of TQFMs is heavily dependent on the choice of loss function. Existing literature lacks a systematic understanding of how different loss functions influence optimization dynamics, computational costs, and classification stability. Specifically, the authors identify a "trajectory crossing problem" in fixed-target training schemes where optimization trajectories intersect, leading to unstable convergence and sharp drops in accuracy.

Methodology
The authors propose a systematic benchmarking framework comparing three distinct loss functions for training TQFMs within a hybrid quantum-classical QSVM pipeline:

1. Distance Loss (DL): Based on the Hilbert-Schmidt distance between mixed density matrices of different classes. This method aims to maximize inter-class separation but requires evaluating all pairwise sample fidelities, resulting in a quadratic computational complexity of $O(M^2)$, where $M$ is the dataset size.
2. Measurement Loss (ML): Assigns each class to a fixed orthogonal basis state and minimizes the error between the encoded state and the target. This approach evaluates samples independently, achieving linear complexity $O(M)$. However, the authors hypothesize that forcing clusters toward fixed targets in a restricted Hilbert space causes trajectory crossings.
3. Log-Likelihood Loss (LLL): The core proposal of this work. Inspired by Maximum Likelihood Estimation, this method reinterprets the fidelity $|\langle \psi | y \rangle|^2$ as a conditional probability $P(y|x)$. Instead of minimizing a linear penalty, LLL minimizes the negative log-likelihood:
   $$L_{LLL}(\theta) = -\frac{1}{L} \sum_{j=1}^{L} \frac{1}{M_j} \sum_{i=1}^{M_j} \log \left( |\langle \psi(x_i^j, \theta) | y_j \rangle|^2 \right)$$
   This formulation applies a non-linear penalty that increases rapidly for samples with low prediction probabilities, thereby emphasizing hard-to-classify samples and providing a stronger optimization signal.

Experimental Setup
The study utilizes the Qiskit state-vector simulator with the EfficientSU2 ansatz. Experiments were conducted on three synthetic datasets (Moons, Circles, Gaussian) representing varying levels of non-linearity and class complexity. The evaluation involved 20 independent runs per configuration to assess statistical stability, varying circuit depths (1 to 6 layers), and testing robustness against simulated depolarizing and readout noise.

Key Results

• Optimization Stability: The ML method exhibited significant instability, characterized by the "trajectory crossing problem" where inter-class overlap spiked and accuracy dropped during early training, particularly in shallow circuits. The LLL method successfully mitigated this, achieving stable convergence comparable to the DL method but without the quadratic cost.
• Computational Efficiency: DL maintained the highest classification accuracy but suffered from $O(M^2)$ scaling, making it impractical for large datasets. Both ML and LLL maintained linear $O(M)$ scaling. The training time for LLL was nearly identical to ML, confirming that the improved stability comes without additional computational overhead.
• Classification Performance:
  • DL consistently achieved the highest mean accuracy across datasets.
  • LLL outperformed ML significantly in terms of stability (lower standard deviation in accuracy across runs) and achieved competitive mean accuracy, often matching or exceeding ML, especially in shallow circuits (e.g., 1-layer circuits on the Circles dataset).
  • Increasing circuit depth (variational layers) helped mitigate trajectory crossing for all methods but introduced a trade-off where excessive depth led to overfitting without proportional gains in test accuracy.
• Noise Robustness: Under simulated noise models, the LLL method maintained robust performance on simpler datasets (Moons) but showed expected degradation on more complex datasets (Circles, Gaussian), consistent with the difficulty of the classification task.

Significance and Claims
The paper claims that the design of the training objective is a critical factor in determining the trainability and generalization of TQFM-based QSVMs. The primary contribution is the introduction of the Log-Likelihood Loss, which offers a practical solution to the trade-off between computational efficiency and optimization stability.

The authors assert that LLL provides a "more stable optimization behavior" than the conventional Measurement Loss while retaining linear computational complexity. This benchmark offers practical guidance for balancing trainability, efficiency, and predictive performance. The work does not claim to solve the fundamental limitations of NISQ devices but rather provides a methodological improvement for the optimization phase of hybrid quantum classifiers.

Future Directions
The authors note that the current framework still relies on a classical SVM for the final decision boundary. They suggest future work should explore fully quantum inference schemes that assign labels directly based on measurement probabilities. Additionally, further investigation is needed into the relationship between circuit expressivity (parameter count vs. structural properties) and optimization performance.