Deterministic generation of a two-dimensional cluster state

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Science  18 Oct 2019:
Vol. 366, Issue 6463, pp. 369-372
DOI: 10.1126/science.aay4354
  • Fig. 1 Scheme of 2D cluster state generation.

    Squeezing is produced by two OPOs (OPOA and OPOB), and coupled into fiber with 97% coupling efficiency. There, temporal modes are interfered with fiber-coupled beam splitters to generate a 2D cluster state. The corresponding graph is shown: Temporal modes of squeezing with mode index k in two spatial modes A and B (bright and dark nodes) are interfered to generate EPR states at BS1. The EPR pairs are entangled to form a 1D cluster state using a τ delay in mode B and BS2, and the 1D cluster state is curled up to a 2D cluster state by another delay of Nτ and BS3. Using homodyne detectors (HDA and HDB), the temporal mode quadratures are measured, from which the nullifiers are calculated. In the experimental implementation, the short delay is a 50.5-m fiber leading to temporal modes of 247-ns duration, whereas the long delay is a 606-m fiber such that N = 12, as shown in the illustrated graph. The temporal modes are defined by an asymmetric-shaped temporal mode function within the 247-ns duration, which filters out low-frequency noise and leads to less than 103 mode overlap (11). For more information, see material and methods (23).

  • Fig. 2 Universality of the generated 2D cluster state.

    (A) Graph of the generated 2D cluster state. Measuring the nodes marked by red in the position basis removes all edges connected to the measured nodes, and the cylindrical graph unfolds to a plane. (B) Resulting plane 2D cluster state after the projective measurements in (A), consisting of two bilayer square lattices (double BSL) connected by edges of weight 1/2. (C) Single BSL after projective measurement of half the modes in (B) in the position basis. (D) Square lattice (SL) after projective position measurements of all modes in spatial mode B (dark nodes), and applying the Fourier gate (π/2 phase delay) on half the modes in spatial mode A (bright nodes). This SL is a traditional universal resource state for MBQC.

  • Fig. 3 Experimental result.

    On the right graph, the nullifiers in Eq. 1 and 2 are shown on the 2D cluster state lattice with the measured variance of 1500 consecutive nullifiers shown in the left plot. Here, the variance is calculated from 10,000 measurements of each nullifier. All nullifier variances are seen to be well below the −3-dB inseparability bound derived in supplementary text section 2 (23), and thus the generated cluster state is completely inseparable. In the inset, the nullifier variance of a larger data set with 2 × 15,000 = 30,000 modes is shown. Again, with all modes below the −3-dB inseparability bound, we conclude the successful generation of a 30,000-mode 2D cluster state. The rapid increase of the variance in n^kx and its periodic variation is caused by phase fluctuation of the squeezing sources, as described in supplementary text section 4 (23).

Supplementary Materials

  • Deterministic generation of a two-dimensional cluster state

    Mikkel V. Larsen, Xueshi Guo, Casper R. Breum, Jonas S. Neergaard-Nielsen, Ulrik L. Andersen

    Materials/Methods, Supplementary Text, Tables, Figures, and/or References

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    • Materials and Methods 
    • Supplementary Text
    • Figs. S1 to S14
    • Table S1 
    • References 

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