2026
Preprint
Fermionic routing
Asymptotically Optimal Depth Fermionic Permutation on 2D Grid Quantum Architecture without Ancillas
Optimal depth scaling.
No extra qubits.
The example is described below.
Each of the sixteen fermionic modes is one strand through the five stages of the compiled circuit on a 4 × 4 grid. Rows and columns exchange places; the two Γ stages correct the relative phase without moving any mode. Copper marks show the operations of the current moment.
The exact 4 × 4 transpose from the public compiler, drawn between its stage boundaries. Spacing in depth shows the five stages; it is not a device layout.
A permutation is also a phase problem.
Route fermionic modes on a 2D grid in optimal O(√N) depth without ancilla qubits.
From the task to an example.
The qubits store fermionic modes. Each mode can be empty or occupied. To run a circuit, we may need to move these modes to new qubits. A square grid permits gates only between neighbours, so the order of these moves matters.
The method runs local gates in parallel. For N qubits, it achieves optimal depth scaling of O(√N) without extra qubits. It preserves the phase signs required for fermions, as well as the final destinations. Getting only the destinations right is not enough.
Try the transpose on the 4 × 4 grid. Mode 1 moves from row 1, column 2 to row 2, column 1. Follow the mode label through the route. During the Γ steps, a stored bit can represent group parity: whether a group has an odd or even number of occupied modes. It does not always represent one mode’s occupation.
The tool uses the fixed 4 × 4 public compiler. It is one example of the construction. The O(√N) result describes how depth scales as the grid grows. See the preprint for the construction and proof.
Follow the modes.
Check the phase.
Choose the destination of each mode. Occupation says which modes hold a particle. A moment is one parallel circuit step.
- 01Row A4moments
- 02Γ · first8moments
- 03Column4moments
- 04Γ · second8moments
- 05Row B3moments
Ancillary qubits are extra helper qubits; this 4 × 4 example uses none. It shows gates, stored bits, mode positions and relative phase. Circuit steps and CNOT-equivalent depth are different counts. The view is a diagram, not a device layout.
Sources &
citation.
arXiv
2026 · Preprint
arXiv:2605.26041 (2026) · Preprint
View the citation
@misc{fermionic-2026,
title = {{Asymptotically Optimal Depth Fermionic Permutation on 2D Grid Quantum Architecture without Ancillas}},
author = {Dantong Li and Shifan Xu and Yongshan Ding},
year = {2026},
eprint = {2605.26041},
archivePrefix = {arXiv},
primaryClass = {quant-ph},
url = {https://arxiv.org/abs/2605.26041}
}