  Spontaneous symmetry breaking phenomena in the transverse field mixed
$p$-spin glass model in finite dimensions are studied with Nishimori's gauge
theory. Useful identities in the gauge theory enable us to study $\mathbb
Z_2$-symmetry breaking. It is proven that ferromagnetic long-range order and
spontaneous magnetization at an arbitrary temperature are bounded by those on
the Nishimori line in the corresponding classical model. These bounds imply
that neither ferromagnetic long-range order nor spontaneous magnetization
exists in spin glass phase. It is proven also that the ferromagnetic
susceptibility has an upper bound in paramagnetic and spin glass phases.
