Core Concepts
Variational Quantum Circuits (VQCs) can be expressed as multidimensional Fourier series, enabling the design of an efficient quantum convolutional neural network architecture for enhanced time series forecasting.
Abstract
The study explores the capabilities of different Variational Quantum Circuit (VQC) architectures and ansatz for time series forecasting, leveraging the theoretical insights that VQCs can be expressed as multidimensional Fourier series.
Key highlights:
Contrary to the common belief that the number of trainable parameters should exceed the degrees of freedom of the Fourier series, the results show that a limited number of parameters can produce Fourier functions of higher degrees, highlighting the remarkable expressive power of quantum circuits.
Reuploading the data into the quantum circuit leads to significantly improved forecasting performance compared to non-reuploading architectures, as it allows the circuit to capture a richer set of Fourier coefficients.
The super-parallel architecture, which reloads the data both vertically and horizontally, outperforms the parallel architecture with an equivalent number of data reloads.
Among the tested ansatz, the Strongly Entangling and Basic Entangler configurations generally yield the best results, with performance improving as the number of qubits increases. However, the Strongly Entangling ansatz exhibits a flatter cost landscape, which can impact training on datasets with fewer samples.
The analysis of Fourier coefficients, expressibility, and gradient variance provides complementary insights into the capabilities and limitations of the different architectures, underscoring the importance of a comprehensive evaluation.
Stats
The time series datasets used in the study include:
Third Legendre polynomial with random noise
Mackey-Glass time series
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Quotes
"Contrary to the condition emphasized in [4] that Np > ν is necessary, our results suggest that a limited number of trainable parameters can yield Fourier functions of higher degrees, underscoring the remarkable expressive power of quantum circuits."
"Employing a super-parallel structure proves more effective than reuploading the data an equivalent number of times in a parallel structure."
"Regarding specific ansatz performances, Strongly Entangler, Custom Entangler, and Basic Entangler consistently yield favorable results, with a tendency for improved metrics as the number of qubits increases."