  The interplay between lattice gauge theories and fermionic matter accounts
for fundamental physical phenomena ranging from the deconfinement of quarks in
particle physics to quantum spin liquid with fractionalized anyons and emergent
gauge structures in condensed matter physics. However, except for certain
limits (for instance large number of flavors of matter fields), analytical
methods can provide few concrete results. Here we show that the problem of
compact $U(1)$ lattice gauge theory coupled to fermionic matter in $(2+1)$D is
possible to access via sign-problem-free quantum Monte Carlo simulations. One
can hence map out the phase diagram as a function of fermion flavors and the
strength of gauge fluctuations. By increasing the coupling constant of the
gauge field, gauge confinement in the form of various spontaneous symmetry
breaking phases such as valence bond solid (VBS) and N\'eel antiferromagnet
emerge. Deconfined phases with algebraic spin and VBS correlation functions are
also observed. Such deconfined phases are an incarnation of exotic states of
matter, $i.e.$ the algebraic spin liquid, which is generally viewed as the
parent state of various quantum phases. The phase transitions between
deconfined and confined phases, as well as that between the different confined
phases provide various manifestations of deconfined quantum criticality. In
particular, for four flavors, $N_f = 4$, our data suggests a continuous quantum
phase transition between the VBS and N\'{e}el order. We also provide
preliminary theoretical analysis for these quantum phase transitions.
