Optimal and efficient qubit routing for quantum simulation

Quantum simulation promises new discoveries in physics, yet on most platforms its progress is limited by device connectivity constraints. Although SWAP gates can be inserted, their use increases circuit depth, which cannot be tolerated on current quantum computers and increases computational cost on fault-tolerant devices. Therefore, minimizing SWAP overhead is crucial, which, however, leads to a computational problem that is itself intractable. We establish a framework for analyzing spatiotemporally periodic circuits, which naturally occur in the quantum simulation of condensed matter systems and lattice gauge theories. We introduce and implement a method that leverages this framework to efficiently minimize SWAP overhead. It has a significant scaling advantage, outperforming leading general-purpose approaches by several orders of magnitude even for moderate system sizes. Remarkably, we find solutions with no SWAP overhead, opening the door for current quantum computers to explore geometrically frustrated magnetism.