On the experimental verification of quantum complexity in linear optics

Journal name:
Nature Photonics
Year published:
Published online


Quantum computers promise to solve certain problems that are forever intractable to classical computers. The first of these devices are likely to tackle bespoke problems suited to their own particular physical capabilities. Sampling the probability distribution from many bosons interfering quantum-mechanically is conjectured to be intractable to a classical computer but solvable with photons in linear optics. However, the complexity of this type of problem means its solution is mathematically unverifiable, so the task of establishing successful operation becomes one of gathering sufficiently convincing circumstantial or experimental evidence. Here, we develop scalable methods to experimentally establish correct operation for this class of computation, which we implement for three, four and five photons in integrated optical circuits, on Hilbert spaces of up to 50,000 dimensions. Our broad approach is practical for all quantum computational architectures where formal verification methods for quantum algorithms are either intractable or unknown.

At a glance


  1. Experimental set-up to generate interfere and detect single photons.
    Figure 1: Experimental set-up to generate interfere and detect single photons.

    ac, Photons generated in a pulsed spontaneous parametric downconversion source are injected, via a V-groove fibre array, to either the QW chip (b) or the RU chip (c). d, Outgoing photons are coupled from the chip using a second fibre array, either directly (not shown) to 16 single-photon avalanche diodes (SPADs), or via a network of fibre splitters (shown). Detection events are recorded with a 16-channel time-correlated single-photon counting system (TCSPC). (See Methods for a detailed description of the experimental set-up and legend abbreviations.)

  2. Three-photon data from the RU chip showing verification of boson sampling  against the uniform distribution  and discrimination between quantum  and classical  statistics.
    Figure 2: Three-photon data from the RU chip showing verification of boson sampling against the uniform distribution and discrimination between quantum and classical statistics.

    a, The expected PDF for values of R* for submatrices chosen from the boson sampling distribution (blue line) and the uniform distribution (black line). The bars show a histogram of R* values from the experimental three-photon data. b, Dynamic updating using the Bayesian model comparison for confidence in sampling from the boson sampling distribution, rather than the uniform distribution. After only 12 threefold detection events we are over 90% confident, and by the end of our experiment we assign only 10−35 probability to the null hypothesis. c, Probability of finding p photons at p detectors (that is, no bunching) for quantum (blue) and classical (red) particles. Lines are theoretical asymptotic values with the constraint m = p2, and histograms (inset) are for theoretically simulated data for up to five photons in 25 modes. Values calculated from our experimental data are shown by the circles in the histograms for three photons in nine modes.

  3. The absence and emergence of multimode correlations in the form of bosonic clouds in three-photon correlation cubes for a nine-mode RU and a 21-mode QW.
    Figure 3: The absence and emergence of multimode correlations in the form of bosonic clouds in three-photon correlation cubes for a nine-mode RU and a 21-mode QW.

    The radii of spheres centred at coordinates (i, j, k) are proportional to the probability of finding three photons in output modes i, j and k, respectively. a,b, We tune between indistinguishable (blue) and distinguishable (red) photons by introducing a time delay between them. These data represent an experimental nine-mode RU with indistinguishable (a) and distinguishable (b) photons. c,d, Bosonic clouds from an experimental 21-mode QW unitary with indistinguishable (c) and distinguishable (d) photons. e,f, Theoretical nine-mode RU with indistinguishable (e) and distinguishable (f) photons. g,h, Theoretical bosonic clouds from 21-mode QW unitary with indistinguishable (g) and distinguishable (h) photons. The experimental data (top row) have been corrected for detector efficiencies and the theory has been filtered to show only events that were experimentally measured, which is the main reason for the apparent asymmetry between the pair of boson clouds.

  4. Experimental verification of correct sampling with bosonic clouding using a QW unitary.
    Figure 4: Experimental verification of correct sampling with bosonic clouding using a QW unitary.

    a, Experimental data for four indistinguishable photons in a 21-mode QW, with black points showing 1,016 of the possible 10,626 detection patterns, ordered by descending theoretical probability (red points). Data circled in blue identify cases of partial or full bunching, which are not included in the clouding metrics here. Error bars are calculated from Poissonian statistics. Here, the input state includes unwanted terms with more than one photon per mode. b, Unwanted parts of the input state are sifted out, so that it approximates to one photon per mode. ce, Results from evaluating our clouding metric for p = 3, 4, 5 photons. Experimental points with horizontal error bars are shown in blue for indistinguishable photons and in red for distinguishable photons; theoretically reconstructed distributions from the same number of samples are shown as solid lines. For three and four photons, the increase in is statistically significant. The separation is reduced for the partially mixed state of five indistinguishable photons across four modes, yet still observable with only 217 counts. In e, the theoretically predicted fall in clouding (blue dotted line) when one of the five photons becomes distinguishable is included. f, Results from the same test for three photons in a nine-mode RU. No significant levels of clouding are observed, as expected, showing that our test is sensitive to the implemented unitary. (See Supplementary Section 3 for further details on data analysis.).


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Author information


  1. Centre for Quantum Photonics, H. H. Wills Physics Laboratory & Department of Electrical and Electronic Engineering, University of Bristol, Merchant Venturers Building, Woodland Road, Bristol, BS8 1UB, UK

    • Jacques Carolan,
    • Jasmin D. A. Meinecke,
    • Peter J. Shadbolt,
    • Nicholas J. Russell,
    • Mark G. Thompson,
    • Jeremy L. O'Brien,
    • Jonathan C. F. Matthews &
    • Anthony Laing
  2. Integrated Optical Microsystems Group, MESA+ Institute for Nanotechnology, University of Twente, Enschede, The Netherlands

    • Nur Ismail &
    • Kerstin Wörhoff
  3. Institute for Mathematical Sciences, Imperial College London, London SW7 2BW, UK

    • Terry Rudolph


Devices were fabricated by N.I. and K.W. All other authors contributed to the theory, experiments, analysis and writing of the manuscript.

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The authors declare no competing financial interests.

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