@techreport{oai:ipsj.ixsq.nii.ac.jp:00217651,
 author = {Ryunosuke, Okubo and Lento, Nagano and Ryunosuke, Okubo and Lento, Nagano},
 issue = {29},
 month = {Mar},
 note = {Variational quantum algorithms (VQAs) are expected to be promising strategies to achieve quantum advantages in the near future. However, gradients of some VQA cost functions vanish exponentially with the number of qubits, which requires exponentially large resources for optimizing them. This phenomenon is the so-called barren plateau problem and has been studied in previous works for certain types of ansatzes. We extend the previous works to a more general type of ansatz. Specifically, we calculate the second moment of a cost function gradient for a general ansatz, assuming that it is an unitary 2-design. We also evaluate the second moment without this assumption, which leads to a relation between a metric to quantify ansatz expressibilities and the second moment. This relation implies cost function landscapes for more expressive ansatzes become flatter. Our results hold independently of ansatz structures, so they are applicable to analysis of scalabilities of various VQAs., Variational quantum algorithms (VQAs) are expected to be promising strategies to achieve quantum advantages in the near future. However, gradients of some VQA cost functions vanish exponentially with the number of qubits, which requires exponentially large resources for optimizing them. This phenomenon is the so-called barren plateau problem and has been studied in previous works for certain types of ansatzes. We extend the previous works to a more general type of ansatz. Specifically, we calculate the second moment of a cost function gradient for a general ansatz, assuming that it is an unitary 2-design. We also evaluate the second moment without this assumption, which leads to a relation between a metric to quantify ansatz expressibilities and the second moment. This relation implies cost function landscapes for more expressive ansatzes become flatter. Our results hold independently of ansatz structures, so they are applicable to analysis of scalabilities of various VQAs.},
 title = {Cost function gradient for general ansatz in variational quantum algorithm},
 year = {2022}
}