ZN gauge theories on a lattice and quantum memory

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 ZN 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. (C) 2003 Elsevier Inc. All rights reserved.