Keywords: quantum neural networks, symmetry, pruning, quantum neural tangent kernel, effective dimension
TL;DR: We prove how the symmetry enhances the training performance of QNNs and then devise an efficient symmetric pruning scheme to distill a symmetric ansatz from an over-parameterized and asymmetric ansatz.
Abstract: Many fundamental properties of a quantum system are captured by its Hamiltonian and ground state. Despite the significance, ground states preparation (GSP) is classically intractable for large-scale Hamiltonians. Quantum neural networks (QNNs), which exert the power of modern quantum machines, have emerged as a leading protocol to conquer this issue. As such, the performance enhancement of QNNs becomes the core in GSP. Empirical evidence showed that QNNs with handcraft symmetric ans\"atze generally experience better trainability than those with asymmetric ans\"atze, while theoretical explanations remain vague. To fill this knowledge gap, here we propose the effective quantum neural tangent kernel (EQNTK) and connect this concept with over-parameterization theory to quantify the convergence of QNNs towards the global optima. We uncover that the advance of symmetric ans\"atze attributes to their large EQNTK value with low effective dimension, which requests few parameters and quantum circuit depth to reach the over-parameterization regime permitting a benign loss landscape and fast convergence. Guided by EQNTK, we further devise a symmetric pruning (SP) scheme to automatically tailor a symmetric ansatz from an over-parameterized and asymmetric one to greatly improve the performance of QNNs when the explicit symmetry information of Hamiltonian is unavailable. Extensive numerical simulations are conducted to validate the analytical results of EQNTK and the effectiveness of SP.
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