  In the present paper we shall study (2+1) dimensional Z_N gauge theories on a
lattice. It is shown that the gauge theories have two phases, one is a Higgs
phase and the other is a confinement phase. We investigate low-energy
excitation modes in the Higgs phase and clarify relationship between the Z_N
gauge theories and Kitaev's model for quantum memory and quantum computations.
Then we study effects of random gauge couplings(RGC) which are identified with
noise and errors in quantum computations by Kitaev's model. By using a duality
transformation, it is shown that time-independent RGC give no significant
effects on the phase structure and the stability of quantum memory and
computations. Then by using the replica methods, we study Z_N gauge theories
with time-dependent RGC and show that nontrivial phase transitions occur by the
RGC.
