  We study the orbital evolution of black hole (BH) binaries in quadruple
systems, where the tertiary binary excites large eccentricity in the BH binary
through Lidov-Kozai (LK) oscillations, causing the binary BHs to merge via
gravitational radiation. For typical BH binaries with masses $m_{1,2}\simeq
20M_\odot-30M_\odot$ and initial semimajor axis $a_0\sim100$ AU (such that the
binaries have no chance of merging by themselves within $\sim10^{10}$ yrs), we
show that binary-binary interactions can significantly increase the LK window
for mergers (the range of companion inclinations that allows the BH binary to
merge within 10~Gyrs). This increase arises from a secular resonance between
the LK oscillation of the BH binary and the nodal precession of the outer
(binary-binary) orbit driven by the tertiary binary. Therefore, in the presence
of tertiary binary, the BH merger fraction is increased to $10-30\%$, an order
of magnitude larger than the merger fraction found in similar triple systems.
Our analysis (with appropriate scalings) can be easily adapted to other
configurations of systems, such as relatively compact BH binaries and
moderately hierarchical triples, which may generate even higher merger
fractions. Since the occurrence rate of stellar quadruples in the galactic
fields is not much smaller than that of stellar triples, our result suggests
that dynamically induced BH mergers in quadruple systems may be an important
channel of producing BH mergers observed by LIGO/VIRGO.
