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

Quantum Physics [Submitted on 15 Apr 2026] Entanglement and circuit complexity in finite-depth random linear optical networks Laura Shou, Joseph T. Iosue, Yu-Xin Wang, Victor Galitski, Alexey V. Gorshkov 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 |\langle\psi|\mathcal U^\dagger \tilde{\mathcal U}|\psi\rangle|^2 for pure states |\psi\rangle with constrained expected photon-number. Comments: 42 pages, 12 figures Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Probability (math.PR) Cite as: arXiv:2604.14277 [quant-ph]   (or arXiv:2604.14277v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.14277 Focus to learn more Submission history From: Laura Shou [view email] [v1] Wed, 15 Apr 2026 18:00:00 UTC (645 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 Change to browse by: math math-ph math.MP math.PR References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)