@techreport{oai:ipsj.ixsq.nii.ac.jp:00211791,
 author = {Kosuke, Ito and Wataru, Mizukami and Keisuke, Fujii and Kosuke, Ito and Wataru, Mizukami and Keisuke, Fujii},
 issue = {16},
 month = {Jun},
 note = {Variational quantum algorithms (VQAs) are expected to become a practical application of near-term noisy quantum computers. Although the effect of the noise crucially determines whether a VQA works or not, the heuristic nature of VQAs makes it difficult to establish analytic theories. Analytic estimations of the impact of the noise are urgent for searching for quantum advantages, as numerical simulations of noisy quantum computers on classical computers are heavy and quite limited to small scale problems. In this work, we establish an analytic estimation of the error in the cost function of VQAs due to the noise. The estimation is applicable to any typical VQAs under the Gaussian noise, which is equivalent to a class of stochastic noise models. Notably, the depolarizing noise is included in this model. As a result, we obtain an estimation of the noise level to guarantee a required precision. Our formulae show how the Hessian of the cost function affects the sensitivity to the noise. This insight implies a trade-off relation between the trainability and the noise resilience of the cost function. As a highlight of the applications of the formula, we propose a quantum error mitigation method which is different from the extrapolation and the probabilistic error cancellation., Variational quantum algorithms (VQAs) are expected to become a practical application of near-term noisy quantum computers. Although the effect of the noise crucially determines whether a VQA works or not, the heuristic nature of VQAs makes it difficult to establish analytic theories. Analytic estimations of the impact of the noise are urgent for searching for quantum advantages, as numerical simulations of noisy quantum computers on classical computers are heavy and quite limited to small scale problems. In this work, we establish an analytic estimation of the error in the cost function of VQAs due to the noise. The estimation is applicable to any typical VQAs under the Gaussian noise, which is equivalent to a class of stochastic noise models. Notably, the depolarizing noise is included in this model. As a result, we obtain an estimation of the noise level to guarantee a required precision. Our formulae show how the Hessian of the cost function affects the sensitivity to the noise. This insight implies a trade-off relation between the trainability and the noise resilience of the cost function. As a highlight of the applications of the formula, we propose a quantum error mitigation method which is different from the extrapolation and the probabilistic error cancellation.},
 title = {Universal noise-precision relations in variational hybrid quantum-classical algorithms},
 year = {2021}
}