SUPPLEMENTARY INFORMATION

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1 DOI: 0.08/NPHOTON.0.0 Experimentl Boson Smpling: Supplementry Informtion Mx Tillmnn,, Borivoje Dkić, René Heilmnn, Stefn Nolte, Alexnder Szmeit, nd Philip Wlther, Fculty of Physics, University of Vienn, Boltzmnngsse, A-090 Vienn, Austri. Institute of Quntum Optics nd Quntum Informtion, Austrin Acdemy of Sciences, Boltzmnngsse, A-090 Vienn, Austri. Institute of Applied Physics, Ae Center of Photonics, Friedrich-Schiller Universität Jen, Mx-Wien-Pltz, D-077 Jen, Germny. CHIP FABRICATION The integrted opticl networks were fricted utilizing femtosecond-lser direct-write technology (see Fig. )). These lser-written wveguides exhiit severl dvntges. Firstly, ritrry wveguide structures cn e written in three dimensions which provides n dditionl degree of freedom for tuning the coupling mong neighoring modes. By tuning this importnt prmeter the polriztion-dependent coupling cn e vried such tht either polriztion-insensitive couplers (e.g. em splitters) or highly polriztion-sensitive couplers (polrizing em splitter) cn e relized. Secondly, the friction process is very fst y requiring.min per wveguide. This llows for short feedck loops, where ech fricted wveguide is tested using microscope nd fluorescence imging vi coherent lser light s well s one- nd two-photon chrcteriztion techniques. Thirdly, the lser-writing cn e done with very high ccurcy of 0nm y using high-precision positioning system (ALS0, Aerotech Inc.). Remrkly this ccurcy is close to the est current photolithogrphic methods tht just reched n ccurcy of out nm []. Friction. The wveguides re written into high-purity fused silic (Corning 7980ArF grde) using regenertive mplifier (RegA 9000, Coherent Inc.) seeded y femtosecond lser system (Mir, Coherent Inc.). The lser-pulse durtion for the write process is 0 fs t 800nm nd the pulse energies re 00nJ t repetition rte of 00kHz. The pulses cn e focused up to 00µm elow the smple surfce using NA = 0.6 ojective. The opticl networks used for the oson smpling computtion re written t depth of 70µm elow the smple surfce, while dditionl structures for lignment nd clirtion procedures re written just 0µm elow the surfce. During the writing process the proe is trnslted with constnt speed of mm/s y high-precision ering stges (ALS0, Aerotech Inc.). In the focus region n effective index chnge of 0 is generted (see Figure ) ). This leds to mode field dimeter for the guided modes of µm µm for wvelength of 789nm (see Fig. ). The mesured coupling with the input fiers (80nm PM-fiers, OZ Optics) ws 6.7dB while losses t the output fcet (grded-index multimode with 0µm core dimeter) were negligile. The propgtion loss ws mesured t 0.dBcm nd the irefringence ws mesured in the order of B = 0 7. Typicl mesured four-fold coincidence count rte ws in the order of min, which is similr to other demonstrtions [, ]. The emplitter rtios nd phses for the x opticl networks were generted rndomly (uniform distriution) on clssicl computer, within the rnge [0, π) for phses nd [0,] for the splitting rtios. In ddition to tht, uncertinties in the writing process of the circuits led to Gussin distriution of the phses nd splitting rtios. Chrcteristics of lser-written networks. The qulity of lser-written wveguides is minly determined y the trnsversl mode profiles nd photon losses over the entire wveguide length. By chieving uniform mode profiles constnt propgtion nd coupling cn e chieved. The high ccurcy of the used positioning system kept the imperfections due to fluctutions in the trnsltion velocity t minimum. However, power fluctutions in the lser system, lthough eing monitored nd controlled, re y fr more dominnt nd led to smll devitions from the desired refrctive index chnge. Since these fluctutions re on shorter time scle thn the required time for writing one wveguide mode, this cn led to devitions from the trgeted device. Through continuous optimiztion we were le to stilize our friction device tht these fluctutions do not cuse mesurle devitions ny more. This cn e seen in Figure, where the theoreticl nd experimentl light propgtion is monitored for stright (Figure nd ) nd curved rry (Figure c nd d) y using fluorescence microscopy technique []. Formlly the coupling constnt etween two wveguides is defined s the overlp integrl [], c n,n+ Ψ n (x)ψ n+ (x)n n+ (x)dx, where Ψ n (x) denotes the electric field in the n th wveguide, nd n n(x) is the squred refrctive index distriution of wveguide. The uncertinty of the refrctive index due to the energy fluctutions of the lser em ws determined to e 0 6, which is two orders of mgnitude elow the verge refrctive index chnge of the wveguides (0 ). Oviously for such smll index fluctutions the mode fields of individul guides remin lmost constnt. The wveguides re fricted in regime where for coupled wveguide modes the coupling constnt is cm. Uncertinties of the refrctive index cuse fluctutions tht re four orders of mgnitude smller thn the coupling constnt nd thus lso of no significnt impct. In Ref. [] the dependence of the coupling constnt on the inter-site spcing ws experimentlly determined to e pproximtely exponentil, resulting in devition of the coupling constnt of lso 0 nd devition of the eting length of 0 cm. The ccording phse devition NATURE PHOTONICS 0 Mcmilln Pulishers Limited. All rights reserved.

DOI: 0.08/NPHOTON.0.0 Figure : ) The writing setup: Femtosecond lser pulses re focused directly into ulk mteril nd crete refrctive index chnge in the focl region.

2 DOI: 0.08/NPHOTON.0.0 Figure : ) The writing setup: Femtosecond lser pulses re focused directly into ulk mteril nd crete refrctive index chnge in the focl region. A high precision positioning system trnsltes the smple, which results in the formtion of wveguide. ) Schemtic of the three-dimensionl writing process. The third dimension ws used to tune the coupling mong neighoring modes. Writing with NA = 0.6 ojective fter slit em shping results in nerly circulr symmetric mode fields with dimeter of µm. is therefore only π/. RECONSTRUCTION To verify the experimentlly otined output distriutions with theoreticl dt, the rndom unitry mtrices hd to e reconstruced. Extrcting the ctul unitry trnsformtion experimentlly [6 8] is importnt s smll imperfections in the friction process cn led to significnt chnges in the trnsformtion mtrix. Friction precision of current stte of the rt technology is fundmentlly limited y resolution of the friction systems. Even for semiconductor device friction this is ounded y the wvelength of the light used in the photolithogrphy process. Only through dvnces in immersion lithogrphy nd doule ptterning smllest process step of nm could e reched. In the cse of fs-written wveguide structures the precision is minly limited y the ccurcy of the positing system. The devices used in this experiment re fricted with positioning precision of 0nm. As the splitting rtio of individul directionl couplers is determined y the interction length nd mode seprtion, which re oth orders of mgnitude igger, this positioning precision is not limiting the friction precision of individul directionl couplers. However it cn led to length fluctutions etween individul elements tht cuse effective phse shifts. For photons with centrl wvelength of 800nm reltive length difference of 0nm etween two modes leds to phse shift of π/8. This ecomes especilly significnt in igger opticl networks consisting of cscded interferometric structures. It is importnt to note, tht this rgument holds independent of the friction method used. This visulizes the importnce of chrcteriztion method to reconstruct the underlying unitry mtrix of the fricted opticl network. We strongly elieve tht chrcteriztion method should e le revel devitions from n initilly trgeted design of n integrted circuit. Only y chrcterizing the contriutions of individul elements to the overll devition of the reconstructed underlying unitry mtrix from the trgeted one efficient optimiztion in the friction process is possile. Rdiometric methods [7] seem to ccomplish tht, ut intrinsiclly rely on loss in the wveguide structures tht limits high performnce opertion. Our chrcteriztion method relies on in situ procedure mening ll mesurements re tken with the circuit lredy plced nd ligned in its finl position. Chrcteriztion mesurements crried out under the exct sme experimentl conditions s the ctul experiment yield the dvntge of minimizing errors due to identicl coupling conditions. We dopt the ide of [8] to utilize single nd two-photon sttes from the downconversion source to perform the chrcteriz- NATURE PHOTONICS 0 Mcmilln Pulishers Limited. All rights reserved.

SUPPLEMENTARY INFORMATION DOI: 0.08/NPHOTON.0.0 c d Figure : ) Experimentl imge of light propgtion in stright lttice. ) The desired propgtion otined y simultions.

The experimentl imges were tken using fluorescence microscopy technique []. tion mesurements.

Opertion on two-photon level ws independently ssured with 0/0 reference directionl coupler lso locted on the chip. Chrcteriztion.

If the structure of n opticl network is known the underlying unitry mtrix cn e expressed in terms of splitting rtios η... η8 nd phse shifts ϕ to ϕ of the individul elements.

3 SUPPLEMENTARY INFORMATION DOI: 0.08/NPHOTON.0.0 c d Figure : ) Experimentl imge of light propgtion in stright lttice. ) The desired propgtion otined y simultions. c) The experimentl imge of light propgtion in curved lttice. d) The desired propgtion otined y simultions. In ll imges the propgtion is top-down. The experimentl imges were tken using fluorescence microscopy technique []. tion mesurements. With tht the circuit is chrcterized for photons of the sme frequency nd spectrl shpe s they re used lter in the experiment. Opertion on two-photon level ws independently ssured with 0/0 reference directionl coupler lso locted on the chip. Chrcteriztion. For the chrcteriztion of the opticl networks nd clcultion of the error rs we utilize mximl likelihood nd Poissonin error propgtion technique (see Ref. [9]). If the structure of n opticl network is known the underlying unitry mtrix cn e expressed in terms of splitting rtios η... η8 nd phse shifts ϕ to ϕ of the individul elements. The ctul vlues of these elements re derived y fitting this unitry to the experimentlly cquired chrcteriztion dt (single photon proilities, two photon visiilities nd reltive output losses). Due to the in situ procedure ll detection efficiencies cn e cquired y injecting single photons in one input mode nd connecting the detectors consecutively to one fier of the multimode output-rry. Coupling losses etween n opticl network nd the output fier rry re negligile due to usge of multimode fiers.thus, the reltive output losses cn e determined experimentlly. Input losses do not ffect the sttistics, s the fitting lgorithm only processes normlized output proilities. It is of considerle dvntge tht two-photon dt s processed y the lgorithm is y nture insensitive to losses. By injecting two indistinguishle photons into two input modes nd tuning the reltive temporl dely etween them non-clssicl interference signl is detected. The visiility of this interference signl for ll possile two fold output comintions is then processed y the fitting lgorithm. By utilizing only single nd two photon sttes Poissonin errors cn e kept negligily smll. Error discussion. To chieve sufficiently high four-fold emission the downconversion source (see Figure ) ws pumped with cw-equivlent power of 700mW. At this power the noise resulting from six-photon emission is still cceptly low. The flse four-fold contriution originting from these six-fold emissions cn e estimted from the chrcteristic prmeters of the source. Tking into ccount the coupling losses nd detection efficiencies s well s the herlding rtio of the source this error contriution ws estimted to e out %. Contriution of systemtic errors from the experiment cn e divided into two prts. The first prt origintes from power fluctutions in the lser system nd smll drifts in the opticl setup tht led to chnge in the genertion rte of the photons nd overll coupling conditions. However, s the three-photon output events stem from three-photon product stte injected into the opticl network, fluctutions in the coupling of the input modes just result in glol chnge in three-photon coincidence count rte. As multimode fiers re used t the output smll fluctutions in coupling hve nerly no effect. Similrly, the power fluctutions in the lser system do not ffect the output distriution ut rther the overll three-photon coincidence count rte. More criticl re fluctutions in the dely lines, s the temporl overlp NATURE PHOTONICS 0 Mcmilln Pulishers Limited. All rights reserved.

4 DOI: 0.08/NPHOTON.0.0 c Proility 0, Proility 0, Proility 0, 0, 0, 0, Figure : Reconstructed unitry mtrices. Asolute vlue squred (proility) of the mtrix elements of ) opticl network no. (mtrix U ), ) opticl network no. (mtrix U ) nd c) opticl network no. (mtrix U ) is plotted. nd therefore quntum interference is ffected. The photons used in this experiment show coherence length of out 90µm nd re ssumed to hve gussin spectrl shpe. With idirectionl repetility of the motorized screws (Newport LTA HL) used in the dely lines of ±µm precision is sufficient with respect to the coherence length of the photons. The ffect of noise nd systemtic errors on the oson smpling computtion is very interesting nd lrgely open question. However there is n indiction tht the influence of noise nd errors for osonic computers is less significnt thn for full-scle universl quntum computers [0]. Reconstructed unitry mtrices. Here we present the reconstructed unitries tht correspond to the opticl networks discussed in the min text. All the errors ( Re(U i ) nd Im(U i )) re evluted ccording to Poissonin distriution. The solute vlue squre of the mtrix elements is plotted in Figure i i i i i i i i 0.80 U = i i i i 0.7, i i i i Re(U ) = , Im(U ) = , () NATURE PHOTONICS 0 Mcmilln Pulishers Limited. All rights reserved.

5 DOI: 0.08/NPHOTON.0.0 SUPPLEMENTARY INFORMATION i i i i i i i i 0.88 U = i i 0. 0.i i 0.8, i i i i Re(U ) = , Im(U ) = i i i i i i i i 0.06 U = i i i i 0.7, i i i i Re(U ) = , Im(U ) = () Fidelity of proility distriutions. We use the stndrd mesure (Bhttchryy distnce) to define the difference etween two proility distriutions. For given p = (p, p...) nd q = (q, q...) the fidelity F( p, q) etween distriutions p nd q is defined s follows: F( p, q) = pi q i. () i [] Intel. World s first nm qud-core processors ring up to twice the visul performnce for unmtched overll pc experiences. Intel Press Relese.0.0 (0). [] Metclf, B. J. et l. Multi-photon quntum interference in multi-port integrted photonic device. rxiv:qunt-ph/08.7 (0). [] Spgnolo, N. et l. Three-photon osonic colescence in n integrted tritter. rxiv:qunt-ph/0.69 (0). [] Szmeit, A. et l. Qusi-incoherent propgtion in wveguide rrys. Applied physics letters 90, (007). [] Szmeit, A., Dreisow, F., Pertsch, T., Nolte, S. & TŘnnermnn, A. Control of directionl evnescent coupling in fs lser written wveguides. Optics Express, (007). [6] Peruzzo, A., Ling, A., Politi, A., Rudolph, T. & O Brien, J. L. Multimode quntum interference of photons in multiport integrted devices. Nture communictions, (0). [7] Thoms-Peter, N. et l. Integrted photonic sensing. rxiv:0.86 (0). [8] Anthony Ling, J. L. O. Super-stle tomogrphy of ny liner opticl device. rxiv: (0). [9] Jmes, D. F. V., Kwit, P. G., Munro, W. J. & White, A. G. Mesurement of quits. Phys. Rev. A 6, 0 (00). [0] Rohde, P. P. & Rlph, T. C. Error tolernce of the oson-smpling model for liner optics quntum computing. Phys. Rev. A 8, 0 (0). URL NATURE PHOTONICS 0 Mcmilln Pulishers Limited. All rights reserved.

Entanglement Purification

Entanglement Purification Lecture Note Entnglement Purifiction Jin-Wei Pn 6.5. Introduction( Both long distnce quntum teleporttion or glol quntum key distriution need to distriute certin supply of pirs of prticles in mximlly entngled