Variational quantum algorithms(VQAs)with random structures have poor trainability due to the exponentially vanishing gradient as the circuit depth and the qubit number increase.This result leads to a general belief th...Variational quantum algorithms(VQAs)with random structures have poor trainability due to the exponentially vanishing gradient as the circuit depth and the qubit number increase.This result leads to a general belief that a deep circuit will not be feasible.In this work,we provide a viable solution to the vanishing gradient problem for deep VQAs with theoretical guarantees.Specifically,we prove that for quantum controlled-layer and quantum residual network(QResNet),architectures,the expectation of the gradient norm can be lower bounded by a value that is independent of the qubit number and the circuit depth.Our results follow from a careful analysis of the gradient behavior on parameter space consisting of rotation angles,as employed in almost all VQAs,instead of relying on impractical 2-design assumptions.We conduct several numerical experiments as verifications,where only our circuits are trainable and converge,while hardware-efficient and random circuits with similar number of parameters in comparison cannot converge.展开更多
基金The national research foundation of Singapore(NRF-P2024-001).
文摘Variational quantum algorithms(VQAs)with random structures have poor trainability due to the exponentially vanishing gradient as the circuit depth and the qubit number increase.This result leads to a general belief that a deep circuit will not be feasible.In this work,we provide a viable solution to the vanishing gradient problem for deep VQAs with theoretical guarantees.Specifically,we prove that for quantum controlled-layer and quantum residual network(QResNet),architectures,the expectation of the gradient norm can be lower bounded by a value that is independent of the qubit number and the circuit depth.Our results follow from a careful analysis of the gradient behavior on parameter space consisting of rotation angles,as employed in almost all VQAs,instead of relying on impractical 2-design assumptions.We conduct several numerical experiments as verifications,where only our circuits are trainable and converge,while hardware-efficient and random circuits with similar number of parameters in comparison cannot converge.
