  Circuit knitting, a method for connecting quantum circuits across multiple
processors to simulate nonlocal quantum operations, is a promising approach for
distributed quantum computing. While various techniques have been developed for
circuit knitting, we uncover fundamental limitations to the scalability of this
technology. We prove that the sampling overhead of circuit knitting is
exponentially lower bounded by the exact entanglement cost of the target
bipartite dynamic, even for asymptotic overhead in the parallel cut regime.
Specifically, we prove that the regularized sampling overhead assisted with
local operations and classical communication (LOCC), of any bipartite quantum
channel is lower bounded by the exponential of its exact entanglement cost
under separable preserving operations. Furthermore, we show that the
regularized sampling overhead for simulating a general bipartite channel via
LOCC is lower bounded by $\kappa$-entanglement and max-Rains information,
providing efficiently computable benchmarks. Our work reveals a profound
connection between virtual quantum information processing via quasi-probability
decomposition and quantum Shannon theory, highlighting the critical role of
entanglement in distributed quantum computing.
