SUPPLEMENTARY INFORMATION

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1 Supplementary Information to Arbitrary integrated multimode interferometers for photonic boson sampling Andrea Crespi, 1, 2 Roberto Osellame, 1, 2, Roberta Ramponi, 1, 2 Daniel J. Brod, 3 Ernesto F. Galvão, 3, Nicolò Spagnolo, 4 Chiara Vitelli, 5, 4 Enrico Maiorino, 4 Paolo Mataloni, 4 and Fabio Sciarrino 4, 1 Istituto di Fotonica e Nanotecnologie, Consiglio Nazionale delle Ricerche (IFN-CNR), Piazza Leonardo da Vinci, 32, I-2133 Milano, Italy 2 Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, I-2133 Milano, Italy 3 Instituto de Física, Universidade Federal Fluminense, Av. Gal. Milton Tavares de Souza s/n, Niterói, RJ, , Brazil 4 Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 5, I-185 Roma, Italy 5 Center of Life La Sapienza, Istituto Italiano di Tecnologia, Viale Regina Elena, 255, I-185 Roma, Italy I. RANDOMLY SAMPLED AND RECONSTRUCTED UNITARIES The matrix U t was sampled from the uniform, Haar distribution over 5 5 unitary matrices: Electronic address: roberto.osellame@polimi.it Electronic address: ernesto@if.uff.br Electronic address: fabio.sciarrino@uniroma1.it ı ı ı ı ı ı ı ı ı U t = ı.87.9ı ı ı ı ı ı ı ı ı, (1).318.9ı ı ı ı ı where the global phase was fixed so as to make U11 t real. We then decomposed U t as a product of matrices that act non-trivially on two modes only, each representing a set of one beam splitter and two phase shifters in the range [,π], as described in [1] and depicted in Fig. 1 of the main text. The use of [,π] phase shifters was an adaptation of the decomposition of [1], which originally used phase shifters in the [, 2π] range. This was done to limit the phase shift (and waveguide deformation) introduced by each element, as this may lead to losses. In this new decomposition, only one of the two phase shifters at each beam splitter input branch needs to introduce a non-zero phase shift. Table I reports the parameters obtained in this decomposition. Note that all experimental outcomes are invariant under multiplication of U t by a phase shifter at each input and output port. For this reason we have set α 1,β 1,β 5,β 8 and β 1 equal to zero in Table I. We have also found a unitary that fits well the singleand two-photon data, in part following the recently proposed method of [2]. In [2], the authors show that the unitary matrix corresponding to a physical interferometer can be reconstructed, up to a round of single phases at the input and output, using only single- and twophoton output probability distributions. The construction requires choosing the matrix elements in one line and one column of the unitary matrix as reference points, and calculating the remaining elements by using a corresponding subset of experimental data consisting of 16 single-photon probabilities and 25 two-photon Hong-Ou- Mandel visibilities. If the physical process is perfectly unitary, this freedom in the choice of the reference elements is irrelevant, as all 25 possible choices produce the same matrix. However, the actual experiment has noise and the data cannot be perfectly described by one unitary matrix. This means that each of the 25 choices of reference elements, each of which corresponding to using one particular subset of the data, produces a slightly different unitary. For our reconstruction, we have obtained these 25 unitaries and calculated how well each of them reproduces the whole data set. The measure of this quality NATURE PHOTONICS 1

2 i t i α i [rad] β i [rad] TABLE I: Transmissivities, t i, and phases, α i and β i, related to the layouts in Fig. 1 of the main text, that result from the decomposition of the sampled unitary matrix. was the χ 2 between the whole set of single- and twophoton output predictions for the unitary with respect to the corresponding experimental data. We then chose the matrix which minimized this measure as a starting point for a numerical optimization. This optimization, a combination of brute force and gradient search methods, slightly modified the best unitary obtained by the previous method to fine-tune the agreement with the experimental data. In our reconstruction algorithm, the non-unitary indistinguishability (q.95) between the two photons has been taken into account. Finally, since the reconstruction method of [2] only obtains the unitary up to a round of arbitrary phases at the input and output modes, we have multiplied each row and column of the optimized unitary so as to obtain the highest gate fidelity with U t, so that the matrices are easier to compare to each other. The reconstructed unitary U r is found to be: ı ı ı ı ı ı ı ı ı U r = ı ı ı ı ı. (2) ı.8.572ı ı ı ı ı.9.524ı ı ı ı Notice that, once again, we have fixed the global phase so as to make U1,1 r real. We have implemented a Monte Carlo simulation to check the consistency of our reconstruction method. We used U r to simulate 1 complete, new data sets for single- and two-photon experiments, with error bars compatible with those of our real experimental data. We then applied the adapted reconstruction method of reference [2] to obtain a reconstructed unitary U r for each data set. The standard deviation in the results serves as our estimated error bars for any quantity of interest. The error bars for the real and for the imaginary parts or U r are respectively: R[U r ]= , (3) and I[U r ]= (4) The similarity between the sampled U t and the reconstructed U r can then be quantified by the gate fidelity F = Tr(U t U r ) /5 =.95 ±.2. Such a value of the gate fidelity F between the sampled unitary U t and the reconstructed one U r has to be related to the fabrication tolerances estimated from the data shown in Fig. 2 of the main text and detailed in the next section. To analyze the effect of fabrication errors in the full device, we sampled N = 1 random unitaries from a gaussian distribution centered around U t. More specifically, the parameters of the internal phase shifters and beam-splitters have been randomly picked from a gaussian distribution centered at the parameter values of U t, with standard deviation equal to the fabrication error. We obtained a value for the average gate fidelity between U t and the sampled unitaries equal to: F =.968 ±.2. Such a value is compatible within one standard deviation with the experimental value of F =.95 ±.2 obtained between U t and U r. II. MODELLING ARBITRARY PHASE SHIFTS AND TRANSMISSIONS Controlled phase shifts between subsequent directional couplers are implemented by varying the optical path, i.e. by stretching the geometrical length of the connection 2 NATURE PHOTONICS

3 waveguide. This is perfomed by applying to the S-bent waveguide path the geometrical transformation described in the following. An undeformed S-bend is described by a sinusoidal function of the kind: y = h ( ) 2π 2 cos L x, (5) where h and L define the extension of the curve in the two coordinates (Fig. 1). Such curve must be deformed and stretched in a smooth fashion, to avoid adding waveguide losses, and without modifying the overall footprint of the S-bend, which would otherwise affect the position of all the other couplers of the network. Hence, the deformation is operated by the following coordinate transformation: ( ) 2π x = x + d sin L x, (6) where d defines the entity of the deformation. The resulting length of the deformed S-bend can be calculated by numerical integration. As discussed and shown in Fig. 2b of the Main Text, the obtained phase shift was experimentally calibrated with respect to the deformation parameter d. A good agreement is observed between the expected phase shift, calculated from the nominal lengthening of the S-bend, and the measured one. The root mean square deviation of the experimentally obtained phase shift with respect to the expected one is.3 rad, equivalent to an error in the path-length of about 25 nm. Independent control on the coupler transmission is obtained, on the other hand, by a three-dimensional rotation of one of the coupler s arm. Indeed, this enables to modify the distance between the two waveguides without altering the shape (and length) of the S-bend, which would determine an unwanted phase shift. To this aim, an accurate modelling of the coupling as a function of the rotation angle is needed. The transmission [6] of a directional coupler, can be expressed by [3]: T = t 2 = sin 2 (κz), (7) where κ is the coupling coefficient and Z is the interaction length (length of the region in which the two waveguides are close to each other) (Fig. 1). The coefficient κ decreases exponentially with the interaction distance s (spacing of the two waveguides within the interaction length) [4]: κ = κ e s s, (8) in which κ and s are suitable constants. These constants have been experimentally measured by fabricating directional couplers in the plane with different s and by fitting the theoretical curve (8) to the experimentally measured transmissions. The values are κ = 42 mm 1 and s =2.4 µm. On the other hand, the distance between the two waveguides depends on the rotation angle α of the arm of the directional coupler according to: s 2 = h 2 +(h + s min ) 2 2h(h + s min ) cos α, (9) where h is defined above and s min is the minimum distance between the waveguides when the two arms are both in the same plane (α = ). Putting together the Equations (7),(8),(9) one reads: ( T = sin 2 ) 1 log κl s e C1 C 2 cos α, (1) with C 1 = h 2 +(h + s min ) 2 and C 2 =2h (h + s min ) for brevity. Thus, inverting the formula (1) one can retrieve the angle α needed for achieving a specific value of T : C 1 s 2 α = arccos ( log κ Z log arcsin T C 2 ) 2. (11) Note that in Fig. 2c of the Main Text the theoretical curve shown in the graph is exactly Eq. (1), thus displaying the excellent agreement of the experimental points with the model here discussed. The root mean square deviation of the experimentally obtained transmission with respect to the expected one is 5%. III. TWO-PHOTON AND THREE-PHOTON MEASUREMENTS L Z Fig. 1: Definition of the relevant parameters in an interferometer node. Scheme of a 3D directional coupler constituting the building block of the 5-mode interferometer, where we define all the relevant geometric parameters. h s α A. Hong-Ou-Mandel visibilities The characterization process is based on performing single-photon and two-photon measurements. The latter consist on injecting all possible combinations of two photons in two different input modes (i, j) and by measuring all possible Hong-Ou-Mandel interferences [5] when the two photons exit in different output ports (K,L). This corresponds to the measurement of a 1 1 visibility matrix, where the visibility is defined as: V 2 (i, j; K, L) = P 2,cl (i, j; K, L) P 2 (i, j; K, L) P 2,cl. (12) (i, j; K, L) NATURE PHOTONICS 3

4 V 2 exp(i, j; K, L) a Here, P 2 (i, j; K, L) is the two-photon probability for inputs (i, j) and output (K, L), and P 2,cl (i, j; K, L) is the two-photon probability when the two photons are delayed out of the interference region. The complete set of experimentally measured visibilities Vexp(i, 2 j; K, L) is shown in Fig. 2a and is compared with the prediction obtained with the reconstructed unitary matrix U r of the interferometer [Fig. 2b]. The agreement between the two matrices is quantified by the similarity Sexp,rp, 2 which is defined as: Sexp,rp 2 i =j K =L =1 V exp(i, 2 j; K, L) Vr,p(i, 2 j; K, L) 2n (13) where n = 1 is the number of measured visibilities. This parameter achieved a value Sexp,rp 2 =.943 ±.3 in our experiment, showing the quality of the reconstruction process. B. Modeling photon distinguishability 1.5!.5! a Pt 3 Pr 3 Pr,p 3 (12) (13)(14) (15) (23)(24) output state (25) (34)(35) (45) (12) (13) (14) (15) (23) (24) (25) (34) (35) (45) input state !.5!1 V 2 r,p(i, j; K, L) (12) (13)(14) (15) (23)(24) output state (25) (34)(35) (45) (12) (13)(14)(15) b (34) (35)(45) (23) (24)(25) input state.5.25 P 3 exp (111) (111) (111) (111) (111) (111) (111) (111) (111) (111) b P 3 r,p Fig. 2: Two-photon Hong-Ou-Mandel visibilities. a, Experimental visibilities Vexp(i, 2 j; K, L) and b, predictions Vr,p(i, 2 j; K, L) with the reconstructed unitary U r obtained with two-photon measurements.,.5 (111) (111) (111) (111) (111) (111) (111) (111) (111) (111) Fig. 3: Three-photon measurements with input state (111). a, Expected three-photon probability distribution for an input state (111): theoretical distribution (Pt 3 ), reconstructed distribution (Pr 3 ) and reconstructed distribution with partial distinguishability (Pr,p). 3 b, Three-photon probability distribution for an input state (111): experimental distribution (Pexp), 3 reconstructed distribution with partial distinguishability (Pr,p). 3 Error bars in the experimental data are due to the Poissonian statistics of the measured events, while error bars on the theoretical predictions have been obtained from a Monte-Carlo simulation. The three-photon state is conditionally prepared by exploiting the second order process of a type-ii parametric down-conversion source. The broadband feature of the pump is responsible for a partial distinguishability be- 4 NATURE PHOTONICS

5 tween the generated photon pairs due to the presence of spectral correlations in the biphoton wave-function. This partial distinguishability introduces a reduction of the interference visibility in the three-photon experiment, and can be modeled by considering an input state of the form: ϱ = r 1,, 1,, 1 1,, 1,, (1 r) 1 a,, 1 b,, 1 a 1 a,, 1 b,, 1 a. (14) Here, r is the parameter which takes into account the indistinguishability between the three photons, and the indexes i = a, b label that the photon on input port 3 belongs to a different photon pair with respect to the photons on input ports 1 and 5. The parameter r is proportional to the overlap between the spectral functions of the three photons, and can be written as r = p 2, where p is the indistinguishability of two photons belonging to different photon pairs. The latter has been measured directly by performing a Hong-Ou-Mandel two-photon interference in a symmetric 5/5 beam-splitter, leading to p =.63 ±.3. C. Additional three-photon measurements In the main text we reported the measured probability distribution for the three-photon input state 1,, 1,, 1. For the sake of completeness, we tested our apparatus by performing the measurement with another three-photon input state randomly picked from all possible choices. The results of the experiment for the, 1, 1, 1, input state are reported in Fig. 3 and compared with the expected predictions with the reconstructed unitary U r and photon indistinguishability p. The values of the variation distance and the similarity between the theoretical prediction and the experiment are reported in the Supplementary Table II. D. Comparison between experiment and theory Single-photon measurements - average over all input states Sexp,r 1 =.99 ±.5 Experiment compared with reconstructed U r d 1 exp,r =.65 ±.3 Sexp,t 1 =.946 ±.5 d 1 exp,t =.158 ±.3 Sexp,r 2 =.977 ±.27 d 2 exp,r =.13 ±.13 Sexp,t 2 =.91 ±.25 d 2 exp,t =.221 ±.13 Experiment compared with theoretical U t Two-photon measurements - average over all input states Experiment compared with reconstructed U r Experiment compared with theoretical U t Three-photon measurements - input state 1,, 1,, 1 S 3 exp,rp =.983 ±.45 Experiment compared with reconstructed U r and photon indistinguishability p d 3 exp,rp =.15 ±.24 S 3 exp,tp =.941 ±.45 Experiment compared with theoretical U t and photon indistinguishability p d 3 exp,tp =.21 ±.24 Three-photon measurements - input state, 1, 1, 1, S 3 exp,rp =.981 ±.44 Experiment compared with reconstructed U r and photon indistinguishability p d 3 exp,rp =.113 ±.22 S 3 exp,tp =.947 ±.42 Experiment compared with theoretical U t and photon indistinguishability p d 3 exp,tp =.174 ±.22 TABLE II: Summary of the experimental data for the single-photon, two-photon and three-photon measurements. For each data set, the experimental distributions are compared with the theoretical prediction obtained from the sampled unitary U t and the one obtained from the actual unitary U r. For each distribution, we report both the similarity S and the variation distance d. The single-photon and the two-photon measurements are averaged over all input state. The experimental data can be compared with the theoretical prediction by adopting two different measures, namely the variation distance and the similarity. The variation distance d between two probability distribu- NATURE PHOTONICS 5

6 tions p i and q i is defined as d = 1 2 i p i q i, and represents the L 1 -norm between the distributions p i and q i. The similarity S between two distributions p i and q i is defined as S =( i pi q i ) 2. Both parameters represent a measure of how close two distributions are (d = and S = 1 for identical distributions). In Table II we report the evaluation of both measures for the experimental data reported in the main text. We evaluated the two quantities by comparing the experimental data with both the predictions obtained by using the theoretical unitary U t and the reconstructed one U r. For the single-photon and the two-photon measurements, the two parameters are evaluated separately for each measured input state and then averaged, while for the three-photon states we report the results for the two measured input states. [1] Reck, M., Zeilinger, A., Bernstein, H. J. & Bertani, P. Experimental realization of any discrete unitary operator. Phys. Rev. Lett. 73, (1994). [2] Laing, A. & O Brien, J. L. Super-stable tomography of any linear optical device. Preprint arxiv: v1 [quantph] (212). [3] A. Yariv. Coupled-mode theory for guided-wave optics. IEEE J. Quantum. Electron. 9, (1973). [4] Szameit, A., Dreisow, F., Pertsch, T., Nolte, S. & Tünnermann, A. Control of directional evanescent coupling in fs laser written waveguides. Opt. Express 15, (27). [5] Hong, C. K., Ou, Z. Y. & Mandel, L. Measurement of subpicosecond time intervals between two photons by interference. Phys. Rev. Lett. 59, (1987). [6] Fraction of power, injected in one waveguide, which is transferred to the other waveguide. 6 NATURE PHOTONICS