Entanglement and circuit complexity in finite-depth random linear optical networks

We study the growth of entanglement and circuit complexity in random passive linear optical networks as a function of the circuit depth. For entanglement dynamics, we start with an initial Gaussian state with all \(n\) modes squeezed. For random brickwall circuits, we show that entanglement, as measured by the R’enyi-2 entropy, grows at most diffusively as a function of the depth. In the other direction, for arbitrary circuit geometries we prove bounds on depths which ensure the average subsystem entanglement reaches within a constant factor of the maximum value in all subsystems, and bounds which ensure closeness of the random linear optical unitary to a Haar random unitary in \(L^2\) Wasserstein distance. We also consider robust circuit complexity for random one-dimensional brickwall circuits, as measured by the minimum number of gates required in any circuit that approximately implements the linear optical unitary. Viewing this as a function of the number of modes and the circuit depth, we show the robust circuit complexity for random one-dimensional brickwall circuits scales at most diffusively in the depth with high probability. The corresponding Gaussian unitary \(tilde{mathcal U}\) for the approximate implementation retains high output fidelity \(|langlepsi|mathcal U^dagger tilde{mathcal U}|psirangle|^2\) for pure states \(|psirangle\) with constrained expected photon-number.