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

• URL: http://arxiv.org/abs/2604.14277v1
• Date: Wed, 15 Apr 2026 18:00:00 GMT
• Title: Entanglement and circuit complexity in finite-depth random linear optical networks
• Abstract summary: We study the growth of entanglement and circuit complexity in random passive linear optical networks as a function of the circuit depth.<n>For random brickwall circuits, we show that entanglement, as measured by the Rényi-2 entropy, grows at most diffusively as a function of the depth.
• Score: 3.387906561638234
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 Rényi-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 $|\langleψ|\mathcal U^\dagger \tilde{\mathcal U}|ψ\rangle|^2$ for pure states $|ψ\rangle$ with constrained expected photon-number.

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