Numerical estimation of the spectal gap of detailed-balanced Lindbladians

The mixing time of a detailed-balanced Lindbladian can be bounded via the spectral gap of the corresponding discriminant superoperator. However, computing the eigenvalues of superoperators presents a significant challenge due to the exponential scaling of their dimension, typically growing as 2^(4L). For one-dimensional local systems, the Density Matrix Renormalization Group (DMRG) is often the most effective method for ground-state estimation. However, modifications to the standard DMRG algorithm are necessary to accommodate the structure of the Gibbs sampler. This talk provides an overview of the methodology used to compute the discriminant gap. We begin with the fundamentals of DMRG with Matrix product States, proceed to the construction of the tensor network representation of the Discriminant, and conclude with the adapted DMRG approach for gap estimation.