SUPPLEMENTARY INFORMATION

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1 Experimental validation of photonic boson sampling - Supplementary Information Nicolò Spagnolo, 1 Chiara Vitelli, 1, 2 Marco Bentivegna, 1 Daniel J. Brod, 3 Andrea Crespi, 4, 5 Fulvio Flamini, 1 Sandro Giacomini, 1 Giorgio Milani, 1 Roberta Ramponi, 4, 5 Paolo Mataloni, 1 Roberto Osellame, 4, 5, Ernesto F. Galvão, 3, and Fabio Sciarrino 1, 1 Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 5, I-185 Roma, Italy 2 Center of Life La Sapienza, Istituto Italiano di Tecnologia, Viale Regina Elena, 255, I-185 Roma, Italy 3 Instituto de Física, Universidade Federal Fluminense, Av. Gal. Milton Tavares de Souza s/n, Niterói, RJ, , Brazil 4 Istituto di Fotonica e Nanotecnologie, Consiglio Nazionale delle Ricerche (IFN-CNR), Piazza Leonardo da Vinci, 32, I-2133 Milano, Italy 5 Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, I-2133 Milano, Italy EXPERIMENTAL APPARATUS, GENERATED STATE AND SIX-PHOTON CONTRIBUTIONS The data acquisition system consisted of a timing circuit driven by the trigger signal and controlled both the conditioning circuit and the four-fold coincidences discrimination and detection circuit. The circuit generates the handshake signal sent to the National PCI-653 board used for ognising the pattern. The four photons injected into the chip were generated by a parametric down-conversion source (shown in Fig. S1) working in a regime in which there is a non-zero probability of generating three photon pairs. The overall state produced up to third order in the parametric gain g is given by: ψ + 2 g ψ g 2 ψ 4 +2g 3 ψ 6, (S1) where ψ 2, ψ 4, and ψ 6 correspond respectively to the two-photon, four-photon and six-photon contributions. In the present implementation, g reaches a value of g.12. The photons are separated in four different spatial modes k i, with i =1,...3, and k T by means of polarization analysis. Photons on mode k T are detected and sent to the detection apparatus, acting as the trigger of the eriment. The other three modes k i, with i =1,...3, are injected into three single mode fiber connected to the integrated devices. The four-photon term is composed by the superposition of three contributions, and is given by the ression ψ 4 =3 1/2 ( 2,,, 2 1, 1, 1, 1 +, 2, 2, ), where n 1,n 2,n 3,n T stands for a Fock state with n i photons on modes k i, with i =1, 2, 3,T. However, due to the polarization analysis before the injection into the single mode fibers, only the 1, 1, 1, 1 term of the generated state can produce a four-fold coincidence event (a trigger photon and three-photon at the output of the interferometer). This polarization analysis excludes at the of the chip the 2,,, 2 and, 2, 2, terms of the generated four-photon state, that would result to the injection ofa(2,, 2) and of (, 2, ) contribution. Due to the spectral correlations of the generated photon pairs, the emitted photons present a degree of partial distinguishability (see Refs. [1]). Furthermore, downconversion sources present a non-zero probability to generate six-photon events. These effects have been taken into account in the ected probability distributions of Fig. 3. The six-photon term ψ 6 = 2 1 ( 3,,, 3 2, 1, 1, 2 + 1, 2, 2, 1, 3, 3, ), generated with a relative probability of 4/3g 2.2 with respect to the four-photon term, can give rise to contributions different from the (1, 1, 1) term. This has been taken into account when considering the theoretical model for the output distribution. Nevertheless, as shown in Ref. [2], such a cortion is small with respect to the main cortion belonging to the three-photon distinguishability. For more details see Refs. [1, 2]. Laser source SHG PDC IF PBS HWP HWP PBS Supplementary Fig. S1: Experimental layout for the generation of the three-photon states. The nm pump beam of the eriment is obtained by loiting the second harmonic generation process (SHG) from a 785 nm pulsed laser source, with duration of 25 fs and average power 7 mw. A parametric down-conversion source (PDC) generates pairs of photons in two different spatial modes, which are spectrally filtered (IF) and selected in polarization by means of an half wave plate (HWP) and a polarizing beam-splitter (PBS). One of the modes k T is measured with a non-photonnumber-resolving avalanche photodiode (APD). The photons generated on the other three modes are injected into the integrated interferometers. k T IF k 1 k 2 k 3 APD NATURE PHOTONICS 1

2 VALIDATION AGAINST THE UNIFORM DISTRIBUTION Here we provide more details on the results reported in Figs. 2a-d of the main text. The erimental results corresponds to the validation protocol for different states. a) Red circle points: modes (2,3,4). b) Four different three-photon combinations. Green circle points: modes (3,4,5); cyan circle points: modes (4,5,6); black circle points: modes (2,4,6); magenta circle points: modes (1,2,3). c) Two different three-photon combinations. Red circle points: modes (4,5,6); black circle points: modes (2,3,4). d) Red circle points: modes (6,7,8). In the reported plots, blue shaded region corresponds to the theoretical prediction of the Boson Sampling validation. More specifically, we simulated a validation eriment over 1 Haar-uniform unitaries, and evaluated the average and the 1.5-standard deviation region for the success probability as a function of the data set size N set. The average is performed by excluding the unitaries where the success rate does not reach 95% even with N set = 5. The number of unitaries with the (asymptotically proven) cort behaviour was respectively 434 (m = 5), 573 (m = 7), 822 (m = 9) and 989 (m = 13), for fixed n, increases with m. Correspondingly, green shaded region corresponds false positives obtained for the validation of a uniform sampler, reported as 1.5-standard deviation average over 1 random Haar-uniform unitaries. To show the test will also work in as-yet unperformed, larger-scale eriments, in Fig. S2 we numerically evaluate the test s success rate for simulated boson sampling eriments with n = 5 and m = 25. Psuccess Boson Sampler EXPERIMENTAL OUTPUT DISTRIBUTIONS Here we provide more erimental output distributions for the 5-, 7-, and 9-mode devices. The results are shown in Figs. S3-S5. More specifically, we sampled one state for the 5-mode device (shown in Fig. S3), four different states for the 7-mode device (shown in Fig. S4), and two states for the 9-mode interferomenter (shown in Fig. 3c of the main text and Fig. S5). To quantify the agreement between the erimental data and the ected distributions, we evaluated the L 1 -norm between the two distributions. The latter distance is defined as d =1/2 k p k q k, being p k and q k the two distributions under analysis. The theoretical prediction are obtained by including the partial photon distinguishability of the generated photons and sixphoton terms in the calculation as discussed in the previous section. For the 5- and 7-mode devices, we performed a full tomographical onstruction of the unitary transformation realized by the implemented interferometers. The adopted method consisted in performing single-photon and two-photon measurements, and by applying a suitable onstruction algorithm [1, 3] to find the unitary transformation which better described the measured data. The gate fidelities F = Tr(U r U t ) /d between the unitaries U r of the implemented device and the desired, ideal unitary matrices U t reached the values F =.95 ±.2 [1] and F =.975 ±.2 for the 5-mode and 7-mode devices respectively. The distance between the erimental distribution and the ected prediction for modes (2,3,4) in the 5-mode device reaches the value d (2,3,4),t =.174 ±.22. For the 7- mode device and the four sampled states, we obtained respectively the distances d (3,4,5),r =.168 ±.8, d (4,5,6),r =.133 ±.17, d (1,2,3),r =.15 ±.7 and d (2,4,6),r =.158 ±.7. In the 9-mode case, we compared ditly the erimentally measured distributions with the predicitions obtained from the ideal unitary. We obtained respectively the values d (4,5,6),t =.113 ±.17 and d (3,4,8),t =.167 ±.2 for the two sampled states, showing the good quality of the fabrication process. Uniform Sampler N set Supplementary Fig. S2: Validation against the uniform distribution. Simulated performance of the validation test for Boson Sampling eriments with n = 5 photons in 1 Haar-uniform 25-mode interferometers, plotting the 1.5- standard deviation average success rate obtained. VALIDATION AGAINST DISTINGUISHABLE SAMPLER Here we provide more erimental data on the validation against the distribution obtained with distinguishable photons. We applied the modified version of the likelihood ratio protocol, discussed in the Methods section, to data sets of different size N set for different states of the 7-mode chip. The ected output probabilities used in the protocol are calculated with respect to the 2 NATURE PHOTONICS

3 Input modes (2,3,4) (125) (134) (135) (145) (234) (235) (245) (345) Supplementary Fig. S3: Distribution for n =3and m =5. Experimental photon number distribution for the 5-mode interferometer with modes (2,3,4). Blue bars: erimental distribution P 3,ind obtained with indistinguishable photons. Light blue region contains the ±σ error interval. Yellow bars: theoretical prediction calculated with the onstructed unitary U r taking into account partial photon distinguishability and six-photon contributions. Light yellow region: contains the ±σ interval taking into account the errors in the unitary onstruction process. fully onstructed unitary U r. The results are shown in Fig. S6. We observe that, the success probability of the validation protocol for indistinguishable photons reaches for all states a value > 9% for a data set size of N set 15. Meanwhile, the probability of obtaining a false positive, that is, the probability of erroneously assigning the data from a Distinguishable Sampler to a Boson Sampler, falls below 2% for the same data set size. We conclude by discussing the numerical simulation of the validation protocols against the Uniform Sampler (Fig. 2e in the main text) and the Distinguishable Sampler (Fig. 4d in the main text), as a function of m and n. In the case of the Aaronson-Arkhipov test against the uniform distribution, we observe that the data set size N min necessary to obtain 95% success rate increases with n (when n and m are small), while it decreases with the number of modes m. Differently, in the case of the modified likelihood ratio test, the data set size N min decreases with n and is independent of the number of modes m. The difference between the two cases is due to the role of multiphoton interference. Indeed, while the P estimator for the Aaronson-Arkhipov algorithm is a quantity related to the output probability of the observed outcome, it does not provide a good approximation for the latter. Thus, an increasing number of photons can result into a larger difference between the coherent, multiphoton behaviour and the row-norm estimator P. On the other hand, multiphoton interference makes more efficient the discrimination between indistinguishable and distinguishable photons for increasing n. Electronic address: roberto.osellame@polimi.it Electronic address: ernesto@if.uff.br Electronic address: fabio.sciarrino@uniroma1.it [1] Crespi, A. et al. Experimental boson sampling in arbitrary integrated photonic circuits. Nature Photonics 7, 545 (213). [2] Spagnolo, N. et al. Three-photon bosonic coalescence in an integrated tritter. Nat. Commun. 4, 166 (213). [3] Laing, A. & O Brien, J. L. Super-stable tomography of any linear optical device. Preprint arxiv: v1 [quantph] (212). NATURE PHOTONICS 3

4 a) Input modes (3,4,5) b) Input modes (4,5,6) Input: c) Input modes (2,4,6) d) Input modes (1,2,3) Supplementary Fig. S4: Distribution for n =3and m =7. Experimental photon number distribution for the 7-mode interferometer with a) modes (3,4,5), b) modes (4,5,6), c) modes (2,4,6) and d) modes (1,2,3). Blue bars: erimental distribution P 3,ind obtained with indistinguishable photons. Light blue region contains the ±σ error interval. Yellow bars: theoretical prediction calculated with the onstructed unitary U r taking into account partial photon distinguishability and six-photon contributions. Light yellow region: contains the ±σ interval taking into account the errors in the unitary onstruction process. 4 NATURE PHOTONICS

5 theo Input modes (3,4,8) (689) (789) Supplementary Fig. S5: Distribution for n =3and m =9. Experimental photon number distribution for the 9-mode interferometer with modes (3,4,8). Blue bars: erimental distribution P 3,ind obtained with indistinguishable photons. Light blue region contains the ±σ error interval. Yellow bars: theoretical prediction calculated with the theoretical unitary U t taking into account partial photon distinguishability and six-photon contributions. 1 Psuccess Boson Sampler Distinguishable Sampler N set Supplementary Fig. S6: Validation against Distinguishable Sampler. Success probability P success of the discrimination protocol with distinguishable or indistinguishable photons as a function of the data set size N set for the 7-mode chip. Green circle points: modes (3,4,5). Blue circle points: modes (2,4,6). Cyan circle points: modes (4,5,6). Magenta circle points: modes (1,2,3). Square points: corresponding success probability of the protocol for the false positive events with distinguishable photons. Error bars are due to the number of collected events. Blue shaded region: numerical simulation of the success probability of discrimination test for indistinguishable photons, taking into account the partial photon distinguishability of the adopted source (see Methods). The results are averaged over 1 Haar-uniform unitaries, and are reported as 1.5-standard deviation. Blue dash-dotted line: average behaviour for perfectly indistinguishable photons. Red shaded region: numerical simulation for distinguishable photons, reported as 1.5- standard deviation, where the partial photon distinguishability of the adopted source has been taken into account in the indistinguishable photon distribution. Red dash-dotted line: average behaviour without taking into account partial photon-distinguishability. NATURE PHOTONICS 5