While quantum machine learning promises to revolutionize data analysis, choosing the most effective quantum circuit known as Ansatz remains a critical issue. Melvin Strobl, M. EmreSahin, Lucas van der Horst, and colleagues at the Karlsruhe Institute of Technology, together with Ben Jaderberg of IBM Quantum, will address this issue by investigating the underlying structures of these circuits. The team reveals that commonly used quantum encoding schemes create predictable relationships between different components of quantum computation and visualize them using “Fourier fingerprints.” This fingerprint offers powerful new ways to identify patterns in random data, reconstruct particle trajectories in high energy physics, optimize quantum machine learning algorithms, and accurately predict how well different Ansatz will work on complex tasks, such as exceeding the limits of existing performance metrics.
Classical data in variation quantum machine learning (QML) leads to quantum Fourier models with numerous Fourier basis functions. Despite this complexity, efficient training requires a limited number of parameters, suggesting that these Fourier modes are not independent, but are correlated with the structure of the quantum circuit itself. Researchers are currently demonstrating this phenomenon and are investigating how these correlations can be used to predict the performance of various quantum circuit designs called unsatisfied.
Fourier coefficient correlation quantifies the training of a circuit
A central finding is that correlations between Fourier coefficients, known as Fourier coefficient correlations (FCC), provide a more reliable measure of quantum circuit performance than traditional metrics such as expressibility and mean square error. Through theoretical analysis, numerical simulations and error analysis, the team demonstrated that the FCC accurately reflects the circuit's ability to learn target functions, demonstrating noise-sensitive and misleading signals. This study establishes the FCC as a robust indicator of the training properties of quantum circuits. Specifically, the results show that FCC is more strongly correlated with actual training performance, especially in complex scenarios.
The metric also shows robustness to errors, shot noise, resulting from limited quantum measurements. The team provides a theoretical foundation that links the FCC to the Fourier representations underlying the target function and the circuit's ability to capture essential functions. Furthermore, FCC has proven to be a more scalable metric than expressiveness, allowing analysis of larger and more complex circuits. Validation using datasets in high energy physics confirms the applicability of FCC to actual problems.
Fourier coefficient correlation limits quantum learning
Researchers found that correlations between Fourier coefficients in quantum feature maps fundamentally affect the performance of quantum machine learning (QML) models. These correlations are because efficient trainable models cannot independently control all the terms in the Fourier series, and cannot effectively limit the complexity of the functions they can learn. The team numerically calculated Fourier coefficient correlations (FCCs) for various quantum circuit designs called “Ansatzes,” and constructed “Fourier fingerprints” to visually represent these correlation structures. The experiments show that FCC accurately predicts the relative performance of different answers when learning random Fourier series, surpassing the widely used “explicitness” metric.
Models showing low FCC values consistently achieve better results, even when explicitness suggests that they are not. This discovery extends to the more complex 2D Fourier series, enhancing the reliability of FCC as a prediction tool. Applying this framework to the challenging problem of jet reconstruction in high energy physics, the team found that Ansatz, with a lower average FCC, also had lower mean quadrature errors, confirming the wider applicability of Fourier fingerprints.
Fourier fingerprints predict the performance of quantum models
This study shows that correlations between Fourier coefficients in quantum feature maps affect the performance of quantum machine learning models. By calculating these correlations and visually representing them as “Fourier fingerprints,” the team improved the performance of both simplified problems and the complex task of jet reconstruction in high energy physics. This suggests that Fourier fingerprints can be a valuable tool for evaluating and predicting the relative performance of various quantum circuit designs called unsatsu. Although the results show a clear trend, the authors acknowledge that the functional maps used produce a limited number of unique frequencies, despite having many basis functions.
This means that completely independent Fourier coefficients are not guaranteed in popular Ansats, and designing the circuit to achieve this remains an open question. Future studies should investigate the relationship between these Fourier coefficient correlations and the training nature of quantum models. These findings should be investigated whether the underlying frequencies extend to other datasets and issues that are not independent. The team proposes that Fourier fingerprints may ultimately act as a guiding bias for incorporating them into quantum feature maps rather than universal performance metrics.
