Topological Modes in Monitored Dynamics

The recent discovery of measurement-induced phase transitions in random quantum circuits has opened up a new avenue to explore non-equilibrium entanglement phase transition driven by the competition between unitary evolution and projective measurements. While prior studies have predominantly studied the critical properties of entanglement phase transitions between volume-law and area-law phases, our work reveals that area-law phases in monitored quantum circuits also manifest nontrivial classifications indicated by their topological properties. This classification is motivated by the one-to-one correspondence between the generic Gaussian quantum circuit and the static noninteracting Hamiltonian in one higher dimension, which is classified by the Altland-Zirnbauer classification. As examples, we explore class DIII and class A circuits whose noninteracting Hamiltonian counterparts are in the same symmetry classes, respectively. In particular, we construct quantum circuits with domain walls separating topologically distinct area-law phases and show that such domain walls carry topologically protected dynamical modes. By designing the domain wall pattern in the circuit, we devise protocols to create, fuse, and even braid these protected dynamical modes. In the case of the class DIII quantum circuit, we find that the protected dynamical modes share the same nature as the Majorana zero modes in p-wave superconductors. Hence, our study offers a novel avenue to explore the fusion and braiding of Majorana modes.