The existence theorem for replica-symmetry breaking (RSB) in the transverse
field Sherrington-Kirkpatrick (SK) model is extended to the model with a
general random exchange interactions. The relation between the expectation
value of the exchange interaction energy and the Duhamel correlation function
of spin operators can be obtained by an approximate integration by parts for
general random interactions. In addition to the Falk-Bruch inequality, these
explicit evaluations enable us to prove that the variance of overlap between
two replica spin operators does not vanish under sufficiently weak transverse
field in sufficiently low temperature. The absence of the ferromagnetic long
range order is also shown to distinguish RSB from the $\mathbb Z_2$-symmetry
breaking.
