Fermions and Colour: Transport and its limits in qutrit–qubit networks

The quantum state of two fermions, such as quarks, changes sign when they are exchanged, and quarks also carry a three-valued internal label called colour. Can networks of small quantum registers reproduce both features? We build each site from four qutrits (three-level quantum systems). Its lowest-energy states record whether the site is occupied and carry a three-component SU(3) colour. Moving the occupation, carrying the colour and obtaining the fermion sign turn out to be separate requirements. Interactions that swap pairs of qutrits move a particle only indirectly, through brief virtual visits to higher-energy states; the same processes also create particle pairs and unwanted colour-dependent returns. A cancellation that removes these on one isolated link fails where links meet, because different links share virtual states. The fermion sign can instead come from preparing the empty sites in an antisymmetric colour state, using a known correspondence between internal symmetry and particle statistics. This needs shared entanglement, allows at most three empty sites, and is not singled out by the hopping Hamiltonian. Adding colour-neutral auxiliary registers keeps the virtual processes on different links apart. Six existing Hamiltonian parameters are then fixed from single-link equations, without fitting masses, mixing angles or propagation scores. At an operating point chosen in advance, a simplified (effective) model of a four-site junction gives minimum fidelity 0.99499 with ideal fermion hopping and maximum leakage out of the encoded states of 0.207%, over sixteen hop times on 102 tested states. A separate implementation reproduces these results. A point with a smaller energy penalty on pair creation fails the same sixteen-hop test. Virtual creation and removal of particle pairs explain, to leading order, the remaining coherent error. A residual calculation bounds the difference between this effective model and the full four-site Hamiltonian. For the same sampled test it gives microscopic fidelity at least 0.99375 and leakage at most 0.326%. The bound is evaluated numerically, without interval certification. This is a finite-model result: the interactions still require supplied three-register controls, precisely calibrated parameters and a prepared background state. Scalable preparation, robustness and a complete microscopic theory of fermions remain open.