Fig. 5: Dual graph and evolution of the iSWAP Gate on dual-rail encoded qubits.
a The graph illustrates the couplings and interactions between Fock states under the Hamiltonian \({\widehat{H}}_{S}\). It consists of 10 vertices, each representing a basis state in the Fock space \({\mathcal{H}}\) of two bosons distributed over four sites. The four selected states span the logical subspace \({{\mathcal{H}}}_{L}\). Blue edges correspond to allowed couplings, red self-loops represent ZZ interactions, and green self-loops correspond to on-site interactions. b1 Coefficient trajectory of the state \({| 0;0\rangle }_{L}\) on the complex plane. The state begins at (1, 0) (green dot) and evolves to (0, 1) (blue dot) at the gate time TS. At the final time, the population of \({| 0;0\rangle }_{L}\) is unity, and the accumulated phase is i. b2 Evolution of the state \({| 0;1\rangle }_{L}\) swapping to \({| 1;0\rangle }_{L}\). The x-axis shows the evolution time in units of TS, while the y-axis indicates the population of the involved states. Despite the non-logical states being populated during the evolution, their populations diminish by the time TS, confirming a perfect swap process.
