Models for nonunitary quantum dynamics, such as quantum circuits that include projective measurements, have recently been shown to exhibit rich quantum critical behavior. There are many complementary perspectives on this behavior. For example, there is a known correspondence between d-dimensional local nonunitary quantum circuits and tensor networks on a [D=(d+1)]-dimensional lattice. Here, we show that in the case of systems of noninteracting fermions, there is furthermore a full correspondence between nonunitary circuits in d spatial dimensions and unitary noninteracting fermion problems with static Hermitian Hamiltonians in D=(d+1) spatial dimensions. This provides a powerful perspective for understanding entanglement phases and critical behavior exhibited by noninteracting circuits. Classifying the symmetries of the corresponding noninteracting Hamiltonian, we show that a large class of random circuits, including the most generic circuits with randomness in space and time, are in correspondence with Hamiltonians with static spatial disorder in the 10 Altland-Zirnbauer symmetry classes. We find the criticality that is known to occur in all of these classes to be the origin of the critical entanglement properties of the corresponding random nonunitary circuit. To exemplify this, we numerically study the quantum states at the boundary of Haar-random Gaussian fermionic tensor networks of dimension D=2 and 3. We show that the most general such tensor network ensemble corresponds to a unitary problem of noninteracting fermions with static disorder in Altland-Zirnbauer symmetry class DIII, which for both D=2 and 3 is known to exhibit a stable critical metallic phase. Tensor networks and corresponding random nonunitary circuits in the other nine Altland-Zirnbauer symmetry classes can be obtained from the DIII case by implementing Clifford algebra extensions for classifying spaces. © 2022 American Physical Society.