Holographic Transformation, Belief Propagation and Loop Calculus for Quantum Information Science

Abstract—The holographic transformation is generalized to problems in quantum information science. In this work, the partition function is represented by an inner product of two highdimensional vectors both of which can be decomposed to tensor products of low-dimensional vectors. On the representation, the holographic transformation is clearly understood by using adjoint linear maps. Furthermore, for the general problem of computing the inner product, the belief propagation and the loop calculus formula are naturally defined and derived, respectively. Finally, an expression of weights in the loop calculus for problems in quantum information science is shown using concepts from quantum information geometry.