Experimental quantum process tomography
This protocol is extracted from research article:
Photonic implementation of Majorana-based Berry phases
Sci Adv, Oct 19, 2018; DOI: 10.1126/sciadv.aat6533

In our experiment, we used the quantum process tomography to identify the efficiency of the performed gate operations (33). The experimental measurement basis is chosen to be {hh, hv, vh, vv}, where h, v, r, and d represent the horizontal, vertical, right-hand circular, and diagonal polarizations, respectively. For each input state, we needed to reconstruct the final output state by two–qubit state tomography with 16 measurement configurations, as shown in fig. S13. To reconstruct the quantum process, we needed 16 different input states. As a result, there are 162 measurement settings. By expanding the output state E(ρ) in terms of the Pauli basis operators {Êm} = {II, IX, IY, IZ, XI, XX, XY, XZ, YI, YX, YY, YZ, ZI, ZX, ZY, ZZ}, the quantum process can be expressed as E(ρ) = ∑mnχmnÊmρÊn. The physical process E is uniquely characterized by the 16-by-16 matrix χ.

In our experiment, the spin basis is represented as Embedded Image, which corresponds to the polarization basis of {hh, hv, vh, vv}. The computation basis is chosen to be {|00g〉, |01g〉, |10g〉, |11g〉}. The transformation between the experimental basis and the computation basis isEmbedded Image(11)The output state in the computation basis can be represented asEmbedded Image(12)whereEmbedded Image(13)A further restriction to the even parity sector can be performed by the projector Pe = (|00g〉〈00g| + |11g〉〈11g|)/2. This results in 4-by-4 reduced density matrices expressed in the logical basis {|00g〉, |11g〉}, as shown in Figs. 3 and 4.

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