Scheme for Universal High-Dimensional Quantum Computation with Linear Optics

Photons are natural carriers of high-dimensional quantum information, and, in principle, can benefit from higher quantum information capacity and noise resilience. However, schemes to generate the resources required for high-dimensional quantum computing have so far been lacking in linear optics. Here, we show how to generate GHZ states in arbitrary dimensions and numbers of photons using linear optical circuits described by Fourier transform matrices. Combining our results with recent schemes for qudit Bell measurements, we show that universal linear optical quantum computing can be performed in arbitrary dimensions.

Figure 1

(a) General schematic for the heralded generation of $N$-photon $d$-dimensional GHZ states via a DFT interferometer. The scheme uses the set of $Nd$ modes $\mathcal{X}$ to encode the qudits and the complementary set of $m-Nd$ modes $\overline{\mathcal{X}}$ are detected for heralding. It requires a single number resolving detector for heralding in the zero-th output mode, while the rest of modes in $\overline{\mathcal{X}}$ use threshold detectors heralding the vacuum. (b) Optimized solution for the heralded generation of Bell states ($N=2$) in arbitrary dimension, requiring $m=2d+1$. The correspondence between the optimal modes and the associated computational states of each qudit is shown. (c) Optimized solution for the heralded generation of three-photon three-dimensional GHZ states using the DFT scheme, requiring $m=25$. Output modes associated to the three different qudits are highlighted with different colors, and the associated qudit computational states are labeled.

Figure 2

Example of modular architecture for constructing universal cluster states from three-photon GHZ states with linear optics. Each module, shown in (a), arranges three-photon GHZ state generators and fusion gates such that a single $d$-dimensional qudit is linked to the four neighboring qudits in the universal square lattice in (b).

Figure 3

Robustness of the qudit Bell state generation circuits against photon indistinguishability. (a) Quantum fidelity of the generated qudit Bell state with the ideal, plotted against photon indistinguishability. (b) The logarithmic negativity bound on the distillable entanglement from the generated state with photon indistinguishability.