Speaker
Description
Geometric quantum machine learning incorporates symmetry into quantum models, using intrinsic problem structure to improve data efficiency, generalization, and trainability. Quantum Fourier models (QFMs) are also important because many angle encoding quantum models have outputs that admit finite Fourier series, so their expressivity can be analyzed through Fourier coefficients. In this work, We study symmetry invariant QFMs. Specifically, we apply an average operator over the chosen symmetry group to obtain an invariant model, and focus on how this symmetry changes the Fourier coefficients. We analyze the variance of symmetrized coefficients under exact and approximate unitary 2-design assumptions on the trainable layers of QFMs. The results show that sufficiently rich symmetry can mitigate vanishing expressivity phenomenon. For approximate 2-designs, our analysis also gives much tighter bounds for general Fourier coefficient variances compared with existing results. We further show that average operators over symmetry groups can realize multivariate Chebyshev polynomial basis functions for PDE learning. To address the high resource cost of implementing the invariant model, we propose randomized encoding, a resource efficient method that implements it by randomly selecting encoding gates. Compared with existing twirling methods, randomized encoding requires no additional qubits or gate depth. To validate the ability of the proposed quantum models to solve PDEs, we implement them as quantum physics-informed neural networks (QPINNs). On two PDE benchmarks in two dimensions, our invariant models clearly outperform QFMs, quantum Chebyshev models, and fully connected PINNs with the same number of parameters.
| Topic | Quantum variational circuits and their applications |
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| Additional Topic | Theoretical foundations of quantum learning |