Recently, we carried out a detailed study of the classical Grote-Hynes theory for chemical reaction rates using a general memory friction kernel. We showed that the singular power-law behavior of the rate near a critical coupling strength and large response times can be described as a critical phenomenon. Consequently, there exists a scaling relation, which reduces the number of exponents needed to describe the rate near the critical point to just one. This exponent depends only on the short-time behavior of the memory friction kernel. Here we study the quantum mechanical Wolynes rate expression from this same viewpoint. The quantum effects turn out to be irrelevant to the principal behavior of the rate. However, corrections to this leading behavior in the critical region can be described by a scaling function, which involves a new exponent. This exponent also depends only on the short-time behavior of the memory friction kernel.