  The time evolution of a physical system is generally described by a
differential equation, which can be solved numerically by adopting a difference
scheme with space-time discretization. This discretization, as a numerical
artifact, results in accumulated errors during evolution thus usually plays a
negative role in simulations. In a quantum circuit, however, the ``evolution
time'' is represented by the depth of the circuit layer, thus is intrinsically
discrete. Hence, the discretization-induced error therein is not a numerical
artifact, but a physical observable effect responsible for remarkable
nonequilibrium phenomena absent in conventional quantum dynamics. In this
paper, we show that the combination of measurement feedback and temporal
discretization can give rise to a new type of quantum dynamics characterized by
unitary but dissipative evolution. As physical consequences of such an unitary
but dissipative evolution, a nonequilibrium steady state with spontaneous
symmetry breaking is revealed in a zero-dimensional (single-qubit) system. A
localization mechanism distinct from that in the well-established Anderson
localization has also been proposed in an one-dimensional interactive quantum
system.
