Asymptotically Optimal Depth Fermionic Permutation on 2D Grid Quantum Architecture without Ancillas

Simulating fermionic systems on qubit hardware involves many nonlocal interactions, and efficient routing of these interactions is critical to the overall cost of fermionic simulation algorithms. Recent works reduce this Jordan-Wigner routing overhead to polylogarithmic depth under all-to-all connectivity, but degrade to O\(sqrt{N} polylog N\) for N fermionic modes on 2D nearest-neighbor architectures. We present a fermionic permutation protocol tailored to 2D grid architectures that achieves the optimal O\(sqrt{N}\) depth with O\(Nsqrt{N}\) nearest-neighbor gates and no ancilla qubits, mid-circuit measurements, or classical feedforward. This matches the Ω\(sqrt{N}\) lower bound, which holds even when O(N) ancillas and classical feedforward are permitted. We further construct an O\(sqrt{N}\)-depth transformation between the Jordan-Wigner, Bravyi-Kitaev, and Parity encodings on the 2D grid via a Hilbert-curve layout, extending our result to all three encodings. Benchmarks on the fermionic fast Fourier transform and Trotter simulation of sparse SYK model demonstrate consistent reduction in depth, spacetime volume, and infidelity for system sizes N gtrsim 100 in the early fault-tolerant regime.