Researchers Reveal Fourier Coefficient Correlations Predict Variational Quantum Circuit Performance with Parameters Scaling As

Quantum machine learning promises to revolutionise data analysis, but selecting the most effective quantum circuit, known as an ansatz, remains a significant challenge. Melvin Strobl, M. Emre Sahin, Lucas van der Horst, and colleagues at the Karlsruhe Institute of Technology, alongside Ben Jaderberg from IBM Quantum, investigate the underlying structure of these circuits to address this problem. The team reveals that commonly used quantum encoding schemes create predictable relationships between different components of the quantum calculation, which they visualise using a ‘Fourier fingerprint’. This fingerprint, they demonstrate, accurately predicts how well different ansatzes perform on complex tasks, such as identifying patterns in random data and reconstructing particle trajectories in high-energy physics, offering a powerful new method for optimising quantum machine learning algorithms and surpassing the limitations of existing performance metrics.

Classical data in variational quantum machine learning (QML) leads to quantum Fourier models with a large number of 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 due to the structure of the quantum circuit itself. Researchers have now demonstrated this phenomenon, exploring how these correlations can be used to predict the performance of different quantum circuit designs, known as ansatzes.

Fourier Coefficient Correlation Quantifies Circuit Trainability

The central finding is that the correlation between Fourier coefficients, termed Fourier Coefficient Correlation (FCC), provides a more reliable measure of quantum circuit performance than traditional metrics like expressibility or mean squared error. Through theoretical analysis, numerical simulations, and error analysis, the team demonstrated that FCC accurately reflects a circuit’s ability to learn a target function, proving less susceptible to noise and misleading signals. This research establishes FCC as a robust indicator of quantum circuit trainability. Specifically, the results show that FCC correlates more strongly with actual training performance, particularly in complex scenarios.

The metric also demonstrates robustness to shot noise, errors arising from limited quantum measurements. The team provides a theoretical foundation linking FCC to the underlying Fourier representation of the target function and the circuit’s ability to capture essential features. Furthermore, FCC proves to be a more scalable metric than expressibility, allowing analysis of larger, more complex circuits. Validation using a dataset from high-energy physics confirms FCC’s applicability to real-world problems.

Fourier Coefficient Correlations Limit Quantum Learning

Researchers have discovered that correlations between the Fourier coefficients of quantum feature maps fundamentally impact the performance of quantum machine learning (QML) models. These correlations arise because efficiently trainable models cannot independently control every term in their Fourier series, effectively limiting the complexity of functions they can learn. The team numerically computed Fourier coefficient correlations (FCCs) for various quantum circuit designs, termed ‘ansatzes’, and constructed a ‘Fourier fingerprint’ to visually represent these correlation structures. Experiments show that the FCC accurately predicts the relative performance of different ansatzes when learning random Fourier series, outperforming the widely-used ‘expressibility’ metric.

Models exhibiting lower FCC values consistently achieve better results, even when expressibility suggests otherwise. This finding extends to more complex two-dimensional Fourier series, reinforcing the reliability of the FCC as a predictive tool. Applying this framework to the challenging problem of jet reconstruction in high-energy physics, the team found that ansatzes with lower average FCC also achieve lower mean squared error, confirming the broader applicability of the Fourier fingerprint.

Fourier Fingerprints Predict Quantum Model Performance

This research demonstrates that correlations between the Fourier coefficients of quantum feature maps impact performance in quantum machine learning models. By calculating these correlations and representing them visually as a ‘Fourier fingerprint’, the team linked lower average correlation to better performance for both simplified problems and the complex task of jet reconstruction in high-energy physics. This suggests the Fourier fingerprint can be a valuable tool for assessing and predicting the relative performance of different quantum circuit designs, known as ansatzes. While the results indicate a clear trend, the authors acknowledge that the feature maps used produce a limited number of unique frequencies despite having many basis functions.

This means that fully independent Fourier coefficients are not guaranteed in popular ansatzes, and designing circuits to achieve this remains an open question. Future research should investigate the relationship between these Fourier coefficient correlations and the trainability of quantum models, as well as explore whether these findings extend to other datasets and problems where the underlying frequencies may not be independent. The team proposes that the Fourier fingerprint may ultimately serve as an inductive bias to incorporate into quantum feature maps, rather than a universal performance metric.