Research ArticleQUANTUM INFORMATION PROCESSING

Experimental scattershot boson sampling

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Science Advances  17 Apr 2015:
Vol. 1, no. 3, e1400255
DOI: 10.1126/sciadv.1400255
  • Fig. 1 Boson sampling and its scattershot configuration.

    (A) Conceptual scheme of boson sampling with n bosons undergoing an arbitrary m-mode unitary transformation. The problem is to sample from the output distribution of the n-bosons over the m-modes. This task can be efficiently performed by a specialized quantum computer performing n-photon interference in an m-mode linear interferometer implementing the chosen unitary transformation. (B) Scattershot configuration for boson sampling with randomly chosen inputs. m heralded single-photon sources, one for each input port, are coupled to the interferometer. During a given time period, n photons (n < m) are probabilistically injected into the interferometer. Each detected n-photon event at the interferometer’s output can be assigned to its corresponding input state by the heralding detectors. Boson sampling is thus performed with random, but heralded, inputs (33, 34).

  • Fig. 2 Experimental layout for the implementation of boson sampling with multiple inputs.

    (A) Overall conceptual scheme of the experiment. (B) In each of the three BBO crystals (Cα, Cβ, and Cγ), photon pairs are generated via type II PDC process. The two possible polarization combinations for the two generated photons, HV and VH, constitute two equal PDC sources enfolded in the same crystal, each one exciting a different trigger (photon V) and a different input mode (photon H). The only exception is given by source S2, whose outputs (I and III in the figure) are both injected in the chip. Sources are also time-multiplexed, because pulses generating photons in crystal Cγ are produced before the ones generating photons in Cα and Cβ. (C) Schematic visualization of the six time- and space-multiplexed photon sources; i is an index of the pump pulse number. (D) For the nine-mode device, the input state is varied manually by changing the input fibers. For the 13-mode device, the input state is varied by the multiple source configuration and by the photon switcher, as described in the main text. Top right inset: Map of the connections between sources and interferometer’s inputs. (E) The photons are then injected into the interferometer by means of a single-mode fiber array and then collected at the output via a multimode fiber array, connected to a set of avalanche photodiodes for detection. (F and G) Internal waveguide design of the 9-mode (F) and 13-mode (G) interferometers. Directional couplers have transmittivity Embedded Image = 0.5, whereas the interferometer’s structure presents static phase shifts with a random pattern. SHG, second harmonic generation; HWP, half wave plate; IF, interference filter; PBS, polarizing beam splitter; APD, avalanche photodiode; PC, polarization controller; SW, fiber switcher.

  • Fig. 3 Multiple input boson sampling in a 9-mode device and scattershot boson sampling in a 13-mode device.

    (A) Density plot of the number ni,j of events detected for each of the 1680 input (i) and output (j) combinations used in our boson sampling experiments with the nine-mode chip. (B) Density plot of the number ni,j of events detected for each of the 2288 input (i) and output (j) combinations used in our scattershot boson sampling experiment with the 13-mode chip. (C and D) Number ni,j of events detected for a two-photon scattershot experiment with the 13-mode chip for input states (9,11) (C) and (11,13) (D).

  • Fig. 4 Validation of multiple-input and scattershot boson sampling against various alternative distributions.

    (A and D) Application of the Aaronson and Arkhipov test against the uniform distribution (A: for the 9-mode chip; D: for the 13-mode chip). (B and E) Application of the likelihood ratio test against distinguishable sampler (B: for the 9-mode chip; E: for the 13-mode chip). (C and F) Success probability Psuccess of the validation protocol against different alternative distributions as a function of the data set size Nset (C: for the 9-mode chip; F: for the 13-mode chip). Horizontal dashed line: 0.95 and 0.05 thresholds for the success probability Psuccess. (A, B, D, and E) Blue points, scattershot boson sampling experimental data; green points, numerical simulation of a uniform sampler; red points, numerical simulation of distinguishable sampler data; dark blue areas, ±2σ region (A and D) or ±1σ region (B and E) expected for the experimental scattershot data, obtained from a numerical simulation, which includes noise in the implemented unitary corresponding to the fabrication tolerances; dark green areas, ±2σ region expected for the uniform sampler; dark red areas, ±1σ region expected for the distinguishable sampler. (C and F) Cyan points, scattershot boson sampling experimental data against the uniform sampler with the Aaronson-Arkhipov test; blue points, scattershot boson sampling experimental data against the distinguishable sampler; orange points, numerical simulation of uniform sampler data against scattershot boson sampler; red points, numerical simulation of distinguishable sampler data against the scattershot boson sampler.

  • Fig. 5 Full simulation of scattershot boson sampling and of its validation.

    (A) Minimum data set size to obtain 95% success probability for the validation of scattershot boson sampling data against the uniform sampler as a function of the number of input states, adopting the Aaronson and Arkhipov test. (B) Minimum data set size to obtain 95% success probability for the validation of a scattershot boson sampling experiment against the distinguishable sampler as a function of the number of input states, adopting the likelihood ratio test. (C) Time required with a laptop to calculate N = 2000 boson sampling probability distributions, each one corresponding to a different input configuration, as a function of the number of modes m, for different number of photons n.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/1/3/e1400255/DC1

    Fig. S1. Characterization of the sources photon indistinguishability by Hong-Ou-Mandel interference in a symmetric beam splitter.

    Fig. S2. Synchronization of single photons belonging to the same source obtained by PDC.

    Fig. S3. Scheme to exploit stimulated emission for the synchronization of photons generated from different crystals.

    Fig. S4. Synchronization of single photons belonging to different sources obtained by amplification of coherent states.

    Fig. S5. Dependence of the variation distance d as a function of the sample size.

    Fig. S6. Contour plot of distribution of pairs [d(data, A), d(data, B)] of variation distances between the output distribution associated with an incorrectly heralded event and either the hypothesis of correct heralded input (A) or the hypothesis of correct input but using distinguishable photons (B).

    Fig. S7. Numerical simulation of a validation test of simulated experimental data against the hypotheses of correct boson sampling data (D > 0), and the hypothesis that photons are distinguishable (D < 0).

    Fig. S8. Ratio Formula between the per-pulse probability of detecting three photons in the trigger apparatus and three photons after the chip without shutters and with shutters, as a function of the number m of sources and modes (g = 0.1, ηT = 0.2, ηD = 0.015).

    References (51, 52)

  • Supplementary Materials

    This PDF file includes:

    • Fig. S1. Characterization of the sources photon indistinguishability by Hong-Ou-Mandel interference in a symmetric beam splitter.
    • Fig. S2. Synchronization of single photons belonging to the same source obtained by PDC.
    • Fig. S3. Scheme to exploit stimulated emission for the synchronization of photons generated from different crystals.
    • Fig. S4. Synchronization of single photons belonging to different sources obtained by amplification of coherent states.
    • Fig. S5. Dependence of the variation distance d as a function of the sample size.
    • Fig. S6. Contour plot of distribution of pairs d(data, A), d(data, B) of variation distances between the output distribution associated with an incorrectly heralded event and either the hypothesis of correct heralded input (A) or the hypothesis of
      correct input but using distinguishable photons (B).
    • Fig. S7. Numerical simulation of a validation test of simulated experimental data against the hypotheses of correct boson sampling data (D > 0), and the hypothesis that photons are distinguishable (D < 0).
    • Fig. S8. Ratio R = Pdet no shutters / Pdet shutters between the per-pulse probability of detecting three photons in the trigger apparatus and three photons after the chip without shutters and with shutters, as a function of the number m of sources and modes (g = 0.1, ηT = 0.2, ηD = 0.015).
    • References ( )

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