Entanglement and circuit complexity in finite-depth random linear optical networks
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arXiv
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| Format: | Preprint |
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2026
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| author | Shou, Laura Iosue, Joseph T. Wang, Yu-Xin Galitski, Victor Gorshkov, Alexey V. |
| author_facet | Shou, Laura Iosue, Joseph T. Wang, Yu-Xin Galitski, Victor Gorshkov, Alexey V. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14277 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Entanglement and circuit complexity in finite-depth random linear optical networks Shou, Laura Iosue, Joseph T. Wang, Yu-Xin Galitski, Victor Gorshkov, Alexey V. Quantum Physics Mathematical Physics Probability 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. |
| title | Entanglement and circuit complexity in finite-depth random linear optical networks |
| topic | Quantum Physics Mathematical Physics Probability |
| url | https://arxiv.org/abs/2604.14277 |