aiqbench analysis
This research introduces the circuit harmonic matrix as a fixed structure for analyzing Fourier expansions in parametrized quantum circuits, explicitly linking design choices like gate types and depth to learning performance. By deriving analytical methods for covariance and tangent kernels, the study reveals that larger coefficient variances correlate with faster learning and identifies spectral biases toward lower frequencies across diverse circuit families. These findings provide a rigorous framework for selecting quantum circuit architectures that are best suited for specific target functions without relying on extensive sampling.
Circuit Harmonic Matrices: A Spectral Framework for Quantum Machine Learning
arXiv:2604.04292v2 Announce Type: replace Abstract: Parametrised quantum circuits learn by adjusting gate parameters, while their design shapes the functions they can represent and how readily they learn them. We introduce the circuit harmonic matrix, a fixed matrix organising Fourier expansions over inputs and parameters. It makes the effects of encoding, gates, initial state and observable explicit in coefficient variance, covariance and the quantum neural tangent kernel, linking variation across parameter space to local sensitivity. The construction also constrains representable functions and lower-bounds fitting error. For fixed Clifford gates and independently parametrised Pauli or controlled rotations, we develop a two-copy propagation method that computes covariance and the averaged tangent kernel analytically for uniform parameters, without constructing the full matrix or sampling parameters. Across 400 configurations spanning eight circuit families, we compare these quantities with actual learning. Larger coefficient variance is strongly associated with faster learning and lower error for the corresponding target frequency, revealing spectral bias towards lower frequencies. Increasing depth and qubit count generally suppresses total variance, yet deeper circuits usually learn faster and achieve lower error, while adding qubits slows early learning and has a frequency-dependent effect on final error. Matched targets show no robust overall advantage from covariance alignment. For one circuit setting, initial tangent kernels closely forecast the modest loss reductions reached in small-step gradient descent. These results connect circuit choices to representational constraints and learning performance, providing a basis for selecting designs suited to particular learning tasks.
Abstract:Parametrised quantum circuits learn by adjusting gate parameters, while their design shapes the functions they can represent and how readily they learn them. We introduce the circuit harmonic matrix, a fixed matrix organising Fourier expansions over inputs and parameters. It makes the effects of encoding, gates, initial state and observable explicit in coefficient variance, covariance and the quantum neural tangent kernel, linking variation across parameter space to local sensitivity. The construction also constrains representable functions and lower-bounds fitting error. For fixed Clifford gates and independently parametrised Pauli or controlled rotations, we develop a two-copy propagation method that computes covariance and the averaged tangent kernel analytically for uniform parameters, without constructing the full matrix or sampling parameters. Across 400 configurations spanning eight circuit families, we compare these quantities with actual learning. Larger coefficient variance is strongly associated with faster learning and lower error for the corresponding target frequency, revealing spectral bias towards lower frequencies. Increasing depth and qubit count generally suppresses total variance, yet deeper circuits usually learn faster and achieve lower error, while adding qubits slows early learning and has a frequency-dependent effect on final error. Matched targets show no robust overall advantage from covariance alignment. For one circuit setting, initial tangent kernels closely forecast the modest loss reductions reached in small-step gradient descent. These results connect circuit choices to representational constraints and learning performance, providing a basis for selecting designs suited to particular learning tasks.
arXiv:2604.04292 [quant-ph]
(or arXiv:2604.04292v2 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2604.04292 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 325 words
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