Figure 1 | Scientific Reports

Figure 1

From: Cyclic permutations for qudits in d dimensions

Figure 1

Cyclic Xd gate. The gate performs the transformation \(|i\rangle \mapsto |i\oplus 1\rangle \) on a qudit. (a) Equivalent quantum network. Spatial modes are used as a qudit ancilla, which is factorised before and after the gate, i.e., starts and ends up in the state |0〉m. There are two SPPs of order +1 and −d (green) and two C(Xd) gates (cyan). The SPP(−d) acts only on mode 0 (open circle on the control mode qudit). (b) Implementation. Photons enter from the left and all OAM states are shifted by SPP(+1). The sorter Sd redirects each state to the corresponding output. The state \(|d{\rangle }_{OAM}\) exits on spatial mode 0 and after the SPP(−d) becomes \(\mathrm{|0}{\rangle }_{OAM}\); all other OAM states \(|j\rangle \), \(j\ne d\), are left invariant. Finally, all states enter the inverse sorter \({S}_{d}^{-1}\) and end up in the same spatial mode |0〉m.

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