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Experimental Scattershot Boson Sampling

Marco Bentivegna, Nicolò Spagnolo, Chiara Vitelli, Fulvio Flamini, Niko Viggianiello, Ludovico Latmiral, Paolo Mataloni, Daniel J. Brod, Ernesto F. Galvão, Andrea Crespi, Roberta Ramponi, Roberto Osellame, Fabio Sciarrino

arXiv:1505.03708v1quant-ph

TL;DR

Experimental Boson Sampling had been limited by the difficulty of scaling to many indistinguishable photons. This paper reports Scattershot experiments using six photon-pair sources and statistical validation; the measured event-rate increase was 4.5 (3.4) relative to standard Boson Sampling, against an expected factor of 5.

  • Problem

    The need for reliable sources of many indistinguishable photons has limited PDC Boson Sampling experiments to three-photon proof-of-principle demonstrations.

    Source: Introduction · pp. 1–2

  • Method

    The experiments connect multiple PDC heralded-photon sources to different interferometer inputs and use statistical tests to validate Scattershot Boson Sampling data.

    Source: Introduction · p. 2

  • Results

    The event rate increased by 4.5 (3.4) versus standard Boson Sampling using a source of average (best) brightness, against an expected factor of 5.

    Source: Discussion · p. 5

  • Takeaways & Limitations

    The experiments demonstrate the feasibility of non-trivial Scattershot realizations using six PDC sources in parallel.

    Source: Discussion · p. 5

    The discussion’s N = 2000 event regime relies on each recorded event being sampled from a different input state with high probability.

    Source: Discussion · p. 5

Abstract

from arXiv · show

Boson Sampling is a computational task strongly believed to be hard for classical computers, but efficiently solvable by orchestrated bosonic interference in a specialised quantum computer. Current experimental schemes, however, are still insufficient for a convincing demonstration of the advantage of quantum over classical computation. A new variation of this task, Scattershot Boson Sampling, leads to an exponential increase in speed of the quantum device, using a larger number of photon sources based on parametric downconversion. This is achieved by having multiple heralded single photons being sent, shot by shot, into different random input ports of the interferometer. Here we report the first Scattershot Boson Sampling experiments, where six different photon-pair sources are coupled to integrated photonic circuits. We employ recently proposed statistical tools to analyse our experimental data, providing strong evidence that our photonic quantum simulator works as expected. This approach represents an important leap toward a convincing experimental demonstration of the quantum computational supremacy.

Introduction

The introduction presents Scattershot Boson Sampling as a way to increase photonic Boson Sampling event rates despite probabilistic photon sources, and reports experiments testing the approach with integrated photonic chips. It frames the work within Boson Sampling’s proposed classical hardness and the challenge of scaling up indistinguishable-photon sources.

Introduction

  • Boson Sampling samples outputs from indistinguishable photons passing through a random interferometer, with classical hardness linked to calculating matrix permanents under mild computational assumptions [2, 3].It is a specialized, non-universal quantum-computing task, and experiments with three photons had been demonstrated in integrated interferometers with up to 13 modes [9] [10] [11] [13] [15].Source: Introduction · p. 1
  • Boson Sampling could support benchmarking and certification of medium-scale devices, while estimates suggest that simulating 30 photons in roughly 100 modes would be extremely demanding classically.Its lack of adaptive measurements means it does not meet the requirements for universal quantum computation, such as efficient integer factoring.Source: Introduction · p. 1
  • Scaling Boson Sampling requires many indistinguishable photons, but PDC sources are probabilistic and their pump power must remain low to limit unwanted higher-order photon-generation terms.PDC offers high indistinguishability and collection efficiency with relatively simple setups, but these constraints had limited demonstrations to three photons.Source: Introduction · pp. 1–2
  • With k > n heralded PDC sources, Scattershot generates n photons in random, heralded input ports with probability ϵ^n, an exponential rate improvement over fixed-input sampling when k ≫ n [33, 34].The sources can be pumped sequentially with little loss to down-converted photons, avoiding a k-fold increase in pump power while keeping higher-order terms low.Source: Introduction · p. 2
  • The experiments use up to six PDC sources on a 13-mode chip for 2- and 3-photon interference, validate it with statistical tests, and also certify results from a 9-mode chip.The authors also numerically discuss simulation and certification complexity and estimate a quantum-supremacy benchmark.Source: Introduction · p. 2

Scattershot Boson Sampling experiment

The experiments implemented Scattershot Boson Sampling in a 13-mode chip using six PDC sources, alongside a preliminary 9-mode experiment. They also demonstrated interference between photons generated by different pump pulses.

  • The preliminary 9-mode experiment used a four-photon PDC source and manually connected 20 input-mode sets to simulate Scattershot Boson Sampling statistics.One photon from the source served as a trigger.Source: Scattershot Boson Sampling experiment · p. 2
  • The 13-mode experiment used six PDC sources, with five supplying random heralded photons and a switcher expanding the possible input sets from five to eight.Two sources were enfolded in each of three crystals; Source S2 injected photons into fixed modes 6 and 8.Source: Scattershot Boson Sampling experiment · p. 3, Figure 2 · p. 3
  • Synchronizing up to six sources across 10 input modes enabled 2- and 3-photon interference; output event maps were sparse, with few or no counts per input-output combination.Controllable relative delays adjusted photon distinguishability; the 9- and 13-mode event maps are shown in Fig. 3.Source: Scattershot Boson Sampling experiment · p. 3, Figure 3 · p. 4
  • The 13-mode chip provided a proof-of-principle demonstration of interference between photons from different pump pulses, using delays to separate the timing of sources S5 and S6.This time-multiplexing approach can increase n-photon generation while using a fixed number of PDC sources.Source: Scattershot Boson Sampling experiment · p. 4

Validation of experimental Boson Sampling data

The section applies statistical tests to partially validate Scattershot Boson Sampling data against alternative sampling hypotheses. It reports successful validation with small data sets while noting that full device certification is computationally hard.

Validation of experimental Boson Sampling data

  • Full certification of correct Boson Sampling device operation is itself computationally hard, so statistical tests provide only partial checks against plausible failure hypotheses.The paper presents Boson Sampling as a test bench for the broader problem of quantum-device certification.Source: Validation of experimental Boson… · p. 4
  • The Aaronson and Arkhipov test [17] uses an efficiently calculated discriminator weakly correlated with Boson Sampling probabilities to test against a uniform output distribution.It was extended from fixed-input to Scattershot data, covering 1,680 input-output combinations on the 9-mode chip and 2,288 on the 13-mode chip.Source: Validation of experimental Boson… · p. 4
  • An adapted likelihood-ratio test compared experimental outcome probabilities for indistinguishable and distinguishable photons, using data combined across input states.The test was applied to both the 9-mode and 13-mode chips.Source: Validation of experimental Boson… · p. 4
  • Successful validation was obtained even with small data sets, with success-rate trends shown for the 9- and 13-mode chips.The success-probability plots compare experimental data against Uniform and Distinguishable Sampler alternatives as data-set size varies.Source: Figure 4 · p. 5, Validation of experimental Boson… · p. 5

Discussion

The first experimental Scattershot Boson Sampling implementation used six parallel PDC sources and increased event rates over standard Boson Sampling. The discussion estimates a substantial runtime advantage while outlining validation conditions, experimental caveats, and routes to scale-up.

Discussion

  • The six-source experiment increased event rates by a factor of 4.5 (3.4) versus standard Boson Sampling with a source of average (best) brightness, compared with an expected factor of 5.The authors attribute the smaller-than-expected increase to non-optimal beam propagation and PDC sources.Source: Discussion · p. 5
  • With N = 2000 events, high probability of sampling a different input state per event requires far more than N input states and one PDC source per input mode.Validation is expected to remain feasible up to about 30 photons because the number of event probabilities requiring classical calculation is low and nearly independent of photon and mode counts.Source: Discussion · pp. 5–6, Figure 5 · p. 6
  • For n = 4 and m = 100, estimated runtime was ∼10^7–10^8 seconds for fixed-input sampling versus ∼50 seconds for Scattershot with k = 100 PDC sources.The estimates use an 80 MHz pulse rate, per-pulse generation probability 0.015, triggering efficiency 0.5, and overall photon-counting probability 0.15.Source: Discussion · pp. 5–6, Figure 5 · p. 6
  • The Fig. 5 simulations omit partial photon distinguishability and other experimental imperfections; incorrectly heralded photons can add white noise and slightly lower validation-test efficiency.A brief optical shutter triggered by the heralding detectors could prevent simultaneous injection of untriggered photons when triggered photons are lost.Source: Discussion · p. 6, Figure 5 · p. 6
  • Scale-up routes include collecting more degenerate type-I PDC modes with microlenses and using time-bin encoding [49].The discussion also identifies other quantum-state classes that preserve Boson Sampling complexity [51] as possible ways to simplify hard-to-simulate device implementations.Source: Discussion · p. 6
  • Further validation proposals examine global coalescence [15], symmetry-based output suppressions [45], or single-mode homodyne detection [44].The authors identify scalable statistical tests against alternative distributions as a useful theoretical direction.Source: Discussion · p. 6

Methods · Fabrication of integrated optics devices

Femtosecond laser direct writing produces buried single-mode waveguides in Eagle2000 glass, which are assembled into 9- and 13-mode integrated interferometers using cascaded directional couplers.

Fabrication of integrated optics devices

Experimental details

Six equal PDC sources in three crystals supplied photons to 9-mode and 13-mode chips, with outputs measured by single-photon detectors and electronic acquisition. The 13-mode chip implemented the complete scattershot experiment using two fixed inputs and a third photon selected shot by shot.

Experimental details

  • The 13-mode chip implemented complete scattershot Boson Sampling with photons injected into fixed input modes 6 and 8 and a third photon supplied shot by shot from one of five remaining PDC sources.The six equal PDC sources were implemented in three crystals.Source: Experimental details · p. 7
  • For the 9-mode chip, a 3-photon input state was generated from the first crystal, using one of four emitted photons as a trigger, and fiber connections manually changed the input modes across 20 configurations.Source: Experimental details · p. 7
  • Multimode fibers connected both chips' outputs to single-photon counting detectors, while electronic acquisition registered events with arbitrary photon numbers.Source: Experimental details · p. 7

Validation of the experimental data

The experimental data are validated against uniform sampling and distinguishable-photon hypotheses using statistical tests. The validation process achieves an average success probability above 95% at the reported dataset sizes.

Validation of the experimental data

  • The uniform-distribution hypothesis is tested with the scalable Aaronson and Arkhipov test [17], experimentally verified in.Source: Validation of the experimental data · p. 7
  • Psuccess > 95% after Nset ∼150 against the uniform distribution and Nset ∼50 against the distinguishable-photon distribution.The same validation procedure was applied to 2-photon data collected simultaneously with the 3-photon data; photons from inputs 11 and 13 were generated from different laser pulses.Source: Validation of the experimental data · p. 7
  • The distinguishability test compares outcome probabilities for indistinguishable and distinguishable photons; the sign of D indicates which hypothesis is likelier, using probabilities calculated by the permanent formula.The calculation accounts for partial photon distinguishability of the source and the chip’s theoretical design parameters.Source: Validation of the experimental data · p. 7
  • For the 9-mode interferometer, separately collected input-state data were mixed uniformly before validation to represent data collected with random input states.Source: Validation of the experimental data · p. 7

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