Criticality and entanglement in non-unitary quantum circuits and tensor networks of non-interacting fermions

There is a correspondence between tensor networks on a (d+ 1)-dimensional lattice and local non-unitary quantum circuits acting on d-dimensional systems. The latter is closely related to circuits composed of unitary evolution and measurements. We show that in the case of non-interacting fermions, there is a further correspondence between non-unitary circuits in d spatial dimensions and non-interacting fermion problems with static Hamiltonians in (d+1) spatial dimensions. Via these correspondences, the non-unitary circuits and their corresponding tensor network can be classified by the generic symmetries of this non-interacting Hamiltonian. Moreover, critical behaviors in random non-unitary quantum circuits can be generically connected to the criticality of non-interacting fermions under quenched disorder. To exemplify this, we numerically study the quantum states at the boundary of Haar-random Gaussian fermionic tensor networks of dimension D= 2 and D= 3. We show that the most general such tensor network ensemble corresponds to a unitary problem of fermions with static disorders in symmetry class DIII, which for both D= 2 and D= 3 is known to exhibit a stable critical metallic phase. Tensor networks in all other symmetry classes can also be systematically constructed.