Feedback-induced interactive dynamics: unitary but dissipative evolution

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.