One of the crown jewels of complexity theory is Valiant's theorem that computing the permanent of an
n
×
n
matrix is
#
P
-hard. Here we show that, by using the model of
linear-optical quantum computing
—and in particular, a universality theorem owing to Knill, Laflamme and Milburn—one can give a different and arguably more intuitive proof of this theorem.
The present work proposes a scheme to teleport a tripartite coherent state using an unparalleled four-component state as a quantum channel. The scheme involves linear optical devices like beam splitters and phase shifters. It is shown that, even by taking an uncompromising quantum model, almost a complete teleportation can be achieved with an impressive number of photons. It is also shown that the teleportation fails only if zero photons are found in all the three output modes or zero in two output modes and a nonzero even/odd photon in one mode. However, the probability of getting these output modes is almost negligible.