Experimental procedure for implementing the ITE of H1
This protocol is extracted from research article:
Photonic implementation of Majorana-based Berry phases
Sci Adv, Oct 19, 2018; DOI: 10.1126/sciadv.aat6533

Consider the eigenvectors Embedded Image, Embedded Image, and Embedded Image of the Pauli operators σx (X), σy (Y), and σz (Z), with eigenvalues {1, − 1}, respectively. Then, the ground state of H0 in Eq. 4 is given byEmbedded Image(5)where α, β, μ, and ν are complex amplitudes satisfying |α|2 + |β|2 + |μ|2 + |ν|2 = 1. Experimentally, the ground state (Eq. 5) of H0 is represented as four spatial modes of single photons, as shown in the initial step of Fig. 2B. To evolve this state to the ground state of H1, we only needed to implement the additional ITE operations of Embedded Image and Embedded Image. Particle 4 is expressed in the basis Embedded Image, as Embedded Image and Embedded Image. This change of basis transformation was implemented by HWPs in the initial four spatial modes. Eight spatial modes were created after splitting them by a BD30. The polarization of the terms with |z4〉, which represent states with higher energy, was set to be vertical with HWPs. The polarization of the terms with Embedded Image was set to be horizontal.

The dissipative evolution was realized by passing photons through a PBS, where only four terms with horizontal polarization remain at the end. Similarly, for the ITE of Embedded Image, the basis of particle 1 was rotated from Embedded Image to Embedded Image with the assistance of a combination of two HWPs and a QWP, as shown in Fig. 2F. Each of the spatial modes was horizontally split into two other modes with a BD30. For particle 2, the basis was rotated from Embedded Image to Embedded Image. The eight spatial modes were further vertically split into 16 modes with a BD60. The terms with the same form were combined with a BD30. Last, the basis of particle 3 was changed to be Embedded Image, in which 16 spatial modes were obtained with another BD30. After passing through a PBS, only the terms Embedded Image, Embedded Image, Embedded Image, Embedded Image, Embedded Image, Embedded Image, Embedded Image and Embedded Image remain, and the output state corresponds to the ground state of H2. The ITE operations of the other Hamiltonians that are part of the cyclic evolution are found in section S1A.

After the basis rotation shown in Fig. 2D, the final state is expressed in the same basis as the initial state and takes the formEmbedded Image(6)To show the gate operation in the logical basis, we translated the basis by Embedded Image, Embedded Image, Embedded Image, and Embedded Image. The logical basis is given by Embedded Image, Embedded Image, Embedded Image, and Embedded Image. The initial state (ground state of H0) is given in the logical basis byEmbedded Image(7)where, for simplicity, we omitted the overall normalization. After the anticlockwise braiding, the final state becomesEmbedded Image(8)The unitary transformation that corresponds to the anticlockwise braiding of MZMs A and C readsEmbedded Image(9)written in the basis {|00g〉, |01g〉, |10g〉, |11g〉}. If we focus on the even fermion parity sector spanned by |00g〉 and |11g〉, the unitary transformation becomesEmbedded Image(10)As a result, the braiding of A and C corresponds to a generalized form of the Hadamard gate operation, related to the standard Hadamard gate by UR2.

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