Fig. 4: Probing fermion dynamics and statistics.

a, System initialized in a two-particle state with complex fermions defined along the ZZ links. An open Pauli string is used to create a fermion pair out of the initial vacuum state. b, Digital evolution under two-qubit interactions in equation (1), realizing Majorana fermion hopping between connected vertices. The hopping strength along the XX and YY links is fixed, J_X = J_Y = J, whereas the value of J_Z/J is tuned. c, Total number of complex fermions in the system during a quench under two different Floquet drives, for J_Z/J = 1 and J_Z/J = 8. The \({K}_{ij}^{{\rm{Z}}}\) term corresponds to the fermion particle number and increasing it enforces particle number conservation. The initial background fermion number resulting from state-preparation errors is subtracted for clarity and the solid lines are the result of noisy numerical simulations (Methods). d, System initialized with two fermions at distances d = 1 or d = 2 is evolved under the Floquet Hamiltonian with J_Z/J = 8. e, A horizontal cut of the two-point density–density correlation relative to the marked site (green dot) at depth 11. The directional asymmetry in hopping is consistent with Pauli exclusion, which prevents fermions from occupying the same site. f, Direct measurement of the fermion exchange phase. A superposition of vacuum and fermion-pair states is created with a sequence of local entangling gates, which realizes a partial-creation unitary R (Methods). The system is evolved under two different particle-hopping protocols: (1) hopping to an intermediate site and hopping back and (2) exchanging the two fermions. The evolution is followed by a second application of R, which maps the exchange phase to fermion presence. Colour plots show the mean fermion density at each site in a 4 × 4 region embedded in the full lattice (Extended Data Fig. 8). g, Contrast between protocols 1 and 2 as a function of decoding postselection. Error bars represent 68% confidence intervals.