  Models for non-unitary quantum dynamics, such as quantum circuits that
include projective measurements, have 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
non-unitary quantum circuits and tensor networks on a D=(d+1)-dimensional
lattice. Here, we show that in the case of systems of non-interacting fermions,
there is furthermore a full correspondence between non-unitary circuits in d
spatial dimensions and unitary non-interacting fermion problems with static
Hermitian Hamiltonians in D=(d+1) spatial dimensions. This provides a powerful
new perspective for understanding entanglement phases and critical behavior
exhibited by non-interacting circuits. Classifying the symmetries of the
corresponding non-interacting 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 ten 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 non-unitary 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 D=3. We
show that the most general such tensor network ensemble corresponds to a
unitary problem of non-interacting fermions with static disorder in
Altland-Zirnbauer symmetry class DIII, which for both D=2 and D=3 is known to
exhibit a stable critical metallic phase. Tensor networks and corresponding
random non-unitary circuits in the other nine Altland-Zirnbauer symmetry
classes can be obtained from the DIII case by implementing Clifford algebra
extensions for classifying spaces.
