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The complex Langevin method and the generalized Lefschetz-thimble method aretwo closely related approaches to the sign problem, which are both based oncomplexification of the original dynamical variables. The former can be viewedas a generalization of the stochastic quantization using the Langevin equation,whereas the latter is a deformation of the integration contour using theso-called holomorphic gradient flow. In order to clarify their relationship, wepropose a formulation which combines the two methods by applying the formermethod to the real variables that parametrize the deformed integration contourin the latter method. Three versions, which differ in the treatment of theresidual sign problem in the latter method, are considered. By applying them toa single-variable model, we find, in particular, that one of the versionsinterpolates the complex Langevin method and the original Lefschetz-thimblemethod.

Combining the complex Langevin method and the generalized Lefschetz-thimble method