Optimizing spontaneous parametric down-conversion sources for boson sampling
R. van der Meer, J.J. Renema, B. Brecht, C. Silberhorn, P.W.H. Pinkse
OOptimizing spontaneous parametric down-conversion sources for boson sampling
R. van der Meer, ∗ J.J. Renema, B. Brecht, C. Silberhorn, and P.W.H. Pinkse COPS, MESA+ Institute for Nanotechnology, University of Twente,PO Box 217, 7500 AE Enschede, The Netherlands Integrated Quantum Optics, Paderborn University,Warburger strasse 100, 33098 Paderborn, Germany (Dated: January 13, 2020)
An important step for photonic quantum technologies is the demonstration of a quan-tum advantage through boson sampling. In order to prevent classical simulability ofboson sampling, the photons need to be almost perfectly identical and almost withoutlosses. These two requirements are connected through spectral filtering, improvingone leads to a decrease of the other. A proven method of generating single photons isspontaneous parametric downconversion (SPDC). We show that an optimal trade-offbetween indistinguishability and losses can always be found for SPDC. We concludethat a 50-photon scattershot boson-sampling experiment using SPDC sources is pos-sible from a computational complexity point of view. To this end, we numericallyoptimize SPDC sources under the regime of weak pumping and with a single spatialmode.
The next milestone in photonic quantum informationprocessing is demonstrating a quantum advantage [1, 2],i.e. an experiment in which a quantum optical sys-tem outperforms a classical supercomputer. This canbe achieved with boson sampling [3]. The aim of bosonsampling is, for a given input configuration of photons,to provide a sample of the output configuration from aarbitraryunitary transformation. A photonic quantumdevice which implements this consists of multiple photonsources, a large passive interferometer and single-photondetectors as shown in Fig. 1. This is believed to be eas-ier to implement than a universal quantum computer andresulted in a surge of experiments [4–11]. These experi-ments require many almost identical photons and prac-tically no losses.Spontaneous parametric downconversion (SPDC)sources are a well-known method of generating singlephotons. A major drawback of building an n -photonSPDC source is the probabilistic generation of the photonpairs, meaning that generating n photons simultaneouslywill take exponentially long. Scattershot boson samplingimproves on this by enabling the generation of n pho-tons in polynomial time using ∼ n sources in parallel[12]. The photons, however, still need to be sufficientlyidentical.A way to improve the photon indistinguishability isspectral filtering. Unfortunately, this comes at the costof losses. Losses, too, are detrimental to multiphotoninterference experiments as they exponentially increasethe experimental runtime [13]. Finding an optimal trade-off between losses and distinguishability is a nontrivialtask.Previous work on optimizing the spectral filteringSPDC sources focused on a trade-off between spectral pu-rity and symmetric heralding efficiency [14]. Other work ∗ [email protected] on designing SPDC sources has studied optimal focusingparameters for bulk crystal sources and pump beam pa-rameters [15], and phase-matching functions [16]. How-ever, the design of SPDC sources specifically for bosonsampling remains an open question as the optimal trade-off between losses and indistinguishability has not beenstudied so far.Recently a new classical approximation algorithm fornoisy boson sampling was suggested which incorporatesboth losses and distinguishability [17]. This algorithmgives a lower bound to the amount of imperfections thatcan be tolerated in order to still achieve a quantum ad-vantage. More importantly, since it incorporates bothimperfections, it can be used to trade-off distinguishabil-ity and losses.In this work, we investigate the design of SPDC sourcesfor scattershot boson sampling from a complexity theorypoint of view. The model of [17] is used to find the op-timal source and filter parameters for a boson-samplingexperiment. From this we determine a minimal overalltransmission efficiency which places a lower bound onthe transmission by other experimental components. Wetarget, by convention, a 50-photon boson-sampling ex-periment [18].Three SPDC crystals are considered: potassium titanylphosphate (ppKTP), β -barium borate (BBO) and potas-sium dihydrogen phosphate (KDP). KTP is a popularchoice since it has symmetric group velocity matching attelecom wavelengths [19, 20], which is favorable for ob-taining pure states. The photon generation rates of KTPsources are high as it uses periodic poling. Moreover,periodic poling allows Gaussian-shaped phase-matchingfunctions by means of Gaussian apodization [21–24]. Thesecond crystal, BBO, is known for generating the currentrecord number of photons [9, 25] and also generates pho-tons at telecom wavelength. However, it has asymmetricgroup velocity matching, resulting in a reduced spectralpurity. Finally, the last crystal we consider is KDP. KDP a r X i v : . [ qu a n t - ph ] J a n E x pe r i m en t H e r a l d n U F FIG. 1. A n -photon scattershot boson-sampling experimenthas n heralded single-photon sources. Each source can senda photon to one of the input modes of the interferometer U .The other photon (dashed) is filtered (F) and is used as aherald. sources, which generate photons at 830 nm, are known togenerate one of the highest purity photons without filter-ing [26].Our calculations consider Gaussian-shaped pulses topump the SPDC process in a collinear configuration. Weassume the existence of only one spatial mode and donot take into account focusing effects. This is a valid as-sumption for both waveguide sources as well as for bulksources without focusing. Focusing increases the numberof spatial modes and hence affects the spectral purity[15]. Furthermore, higher-photon-number states are ig-nored, which is reasonable given the existence of photon-number-resolving detectors [27]. I. THEORYA. SPDC sources
SPDC sources turn a pump photon into two down-converted photons, and hence produce photons in pairs.For Type-II SPDC, the two photons from the pair eachemerge in a separate mode. Traditionally these modesare referred to as signal and idler. The SPDC processcan be understood by considering energy conservation¯ hω p = ¯ hω s + ¯ hω i as well as momentum conservation (cid:126)k p = (cid:126)k s + (cid:126)k i , where p, s and i denote the pump, signal andidler photons, respectively. Momentum conservation canbe tweaked by quasi phase matching by either periodicor apodized poling. Both energy and momentum conser-vation only allow certain wavelength combinations andtogether they specify the spectral-temporal properties ofthe two-photon state [28].Birefringence results in an asymmetry between the sig-nal and idler photon. This leads to spectral-temporal I d l e r W a v e l eng t h Signal Wavelength 01 pha s e m a t c h i ng ene r g y c on s e r v a t i on N o r m a li z ed I n t en s i t y FIG. 2. An example of a joint spectral intensity (JSI). Thered dashed line shows the Gaussian filter for both the signaland idler photon. The (anti)diagonal white lines denote theregion which satisfies phase matching (energy conservation). correlations between the two. Such correlations reducethe spectral purity P x = Tr( ρ ), where ρ x is the reduceddensity matrix of photon x. When no correlations ex-ist, the photon state is factorizable and the photons arespectrally pure [29].A visual representation of the two-photon state isshown in Fig. 2. The spot in the center indicates thatthe two-photon state with what probability the photonsare in this region of the frequency space. This probabil-ity is also referred to as the joint spectral intensity (JSI),which is related to the joint spectral amplitude (JSA) byJSI = | JSA | . The JSA describes the wavefunction ofthe photon pair as a function of the wavelength of thephotons and follows from energy and momentum con-servation. The factorizability of the JSA determines thespectral purity of the source.We now proceed with a mathematical description ofthe JSA, which follows from energy and momentum con-servation. The energy conservation α ( ω s , ω i ) function isa Gaussian pulse with a center wavelength ω p and band-width σ p : α ( ω s , ω i ) = exp (cid:18) − ( ω s + ω i − ω p ) σ (cid:19) . (1)The phase-matching function for a periodically poledcrystal is given by: φ ( ω s , ω i ) = sinc (cid:18) k p − k s − k i − π Λ L (cid:19) , (2)with L the length of the nonlinear crystal and Λ thepoling period. Another type of quasi phase matchingexists, which is Gaussian apodization [21] φ G ( ω s , ω i ) = exp (cid:18) − γ ∆ k L (cid:19) , (3)where γ ≈ . k denotes the phase mismatch and L again the crystallength. The energy conservation function together withthe appropriate phase-matching function give the JSA: f ( ω s , ω i ) = α ( ω s , ω i ) φ ( ω s , ω i ) . (4)The two-photon state corresponding to this JSA can giverise to distinguishability. This can be mitigated by spec-tral filtering. The overall two-photon state after filteringcan now be written as: | ψ (cid:105) = (cid:90) (cid:90) dω s dω i f ( ω s , ω i ) F s , i ( ω s , ω i ) | s (cid:105)| i (cid:105) , (5)where F s , i ( ω s , ω i ) denotes a possible filter function on thesignal and/or idler photon. For simplicity, we ignore thevacuum and multiphoton states.The spectral purity of the photon pair can be foundwith a Schmidt decomposition of the JSA [30, 31]. Fromthis follows a Schmidt number K which determines thespectral purity P = 1 K . (6)Physically, K is the effective number of modes that isrequired to describe the JSA (e.g., see [32]). When K = 1the photon pair is factorizable. In this case, detecting aphoton as herald leaves the other photon in a pure state.In Fig. 2 this would manifest itself such that the JSAbecomes aligned with the axes. In case K >
1, detectingone photon leaves the other photon in a mixed state ofseveral modes. Hence, the remaining photon has a lowerspectral purity.It is possible to improve the spectral purity by filteringthe photons. The effect of filtering can be understood asoverlaying the filter function over the JSA. This is shownwith the dashed lines in Fig. 2. A well-chosen filter re-moves the frequency correlations between the photons,but inevitably introduces losses, which in turn are detri-mental for boson-sampling experiments.
B. Classical simulation of boson sampling withimperfections
The presence of imperfections such as losses [33] anddistinguishability [34] of photons reduces the computa-tional complexity of boson sampling. Classical simula-tion algorithms of boson sampling upper bound the al-lowed imperfections. These classical simulations approx-imate the boson sampler outcome with a given error.We now present the model of [17]. This model ap-proximates an imperfect n -photon boson sampler where n − m photons are lost, by describing the output as up to k -photon quantum interference (0 ≤ k ≤ m ) and at least m − k classical boson interference. Furthermore, this for-malism naturally combines losses and distinguishability into a single simulation strategy, thereby introducing anexplicit trade-off between the two. In this model, theerror bound E of the classical approximation is given by E < (cid:115) α k +1 − α . (7)The parameter α which we will refer to as the ’sourcequality’ is given by α = ηx , (8)with η = m/n denotes the transmission efficiency perphoton. Losses in different components are equivalent,so different losses can be combined into a single param-eter η [35]. The average overlap of the internal partof the wave function between two photons is given by x = (cid:104) ψ i | ψ j (cid:105) (i (cid:54) =j). Therefore x is the visibility of asignal-signal Hong-Ou-Mandel interference dip [36]. Thisindistinguishability equals the spectral purity.This model allows for optimizing the SPDC configura-tion by optimizing the source quality of Eq. 8, which ef-fectively trades-off the losses and distinguishability. Fur-thermore, from Eq. 7 the maximal number of photons k can be calculated by specifying a desired error bound. II. METHODS
In order to find the best SPDC configuration for a se-lection of crystals, we run an optimization over the SPDCsettings to maximize α while varying the filter band-width. Since we consider collinear SPDC, the optimiza-tion parameters are the crystal length L and the pumpbandwidth σ p . Note that these parameters determinethe shape of the JSA and therefore the separability. Thepump center wavelength is set such that group velocitydispersion is matched [26, 37–39]. From our numericalcalculations we observe that the optimization problemappears to be convex over the region of the parameterspace of interest. We note that the optimization param-eters are bounded, e.g., the crystal length cannot be neg-ative. A local optimization routine (L-BFGS-B, Python)was used.The source quality α can be calculated from the JSA.The JSA was calculated numerically by discretizing thewavelength range of interest [40]. The wavelength rangewas chosen to include possible side lobes of the sincphase-matching function. The spectral purity is calcu-lated from the discretized JSA using a singular value de-composition (SVD) [41]. The transmission efficiency iscalculated by the overlap of the filtered and unfilteredJSA. In other words, only ’intrinsic’ losses are consid-ered and experimental limitations such as additional ab-sorption by optical components or absorption losses inthe crystal are not taken into account. This is permissi-ble since such experimental losses are constant over thewavelength range. P ho t on T r an s m i ss i on S ou r c e Q ua li t y KDPKDP R.ppKTPppKTP R. apKTPBBOBBO R. w ea k fi l t e r i ng s t r ong fi l t e r i ng k = I n t e r f e r ab l e P ho t on s k k = a) b) k = Photon Indistinguishability x Filter Bandwidth FWHM [nm]
FIG. 3. a) The transmission efficiency per photon η and indistinguishability x corresponding to the ideal SPDC settings atdifferent filter bandwidths for different crystals (see legend in b). The dashed lines are isolines, indicating how many photons k can be used for a boson-sampling experiment. The indistinguishability and transmission efficiency together result in the sourcequality factor α = x η . b) The values of α and the corresponding number of photons k (right axis) as function of the filterbandwidth. In the legend R. denotes a rectangular filter, otherwise a Gaussian filter was used. The introduction of wavelength-independent lossesdoes not chance the position of the optimum, as it only re-duces the transmission efficiency. Wavelength-dependentlosses can be understood as an additional filter.Realistic SPDC settings are guaranteed by constrain-ing the crystal length and pump bandwidth values in theoptimizer. The crystal lengths are bounded by what iscurrently commercially available. The pump bandwidthis bounded to a maximum of roughly 25 fs (∆ f ≈
17 THz)pulses. Such pulses can be realized with commercialTi:Sapph oscillators. See the supplementary materialsfor the exact bounds and further details. Furthermore weconsider Gaussian-shaped and rectangular-shaped band-pass filters. Rectangular filters are a reasonable approx-imation of broadband bandpass filters.In the calculations, only the herald photon is filtered.Also filtering the other photon reduces the heralding ef-ficiency. Typically the increase in purity is not worththe additional losses, especially if finite transmission ef-ficiency of filters is included.
III. RESULTS
We now proceed by using the metric of [17] to computethe optimal filter bandwidth, pump bandwidth and crys-tal length for KTP, BBO and KDP sources. The upperbound for the error of the classical approximation (Eq.7) is set on the conventional E = 0 . α . The transmission efficiency η is shown on the y-axisand signal-signal photon indistinguishability x on the x- axis. The ideal boson-sampling experiment is located atthe top right. Each point represents an optimal SPDCconfiguration that maximizes α for that crystal corre-sponding to a fixed filter bandwidth. The black dashedisolines indicate the maximum number of photons k onecan interfere, i.e., they are solutions of Eq. 7 for a fixed E and α . The weak-filtering regime is in the top left, andthe strong-filtering regime is in the bottom right.Figure 3b) represents the source quality α from Fig.3a) explicitly as a function of the filter bandwidth. Theleft y-axis indicates the source quality α . The right y-axis shows the corresponding maximal number of photons k . Both graphs show that there is a filter bandwidththat maximizes α . From this maximal α opt the minimaltransmission budget η TB can be defined η TB α opt = α , (9)where α denotes the required value of α to perform a50-photon boson-sampling experiment. The transmissionbudget defines the minimum required transmission effi-ciency for all other components together. This includes,for instance, non-unity detector efficiencies. The maxi-mal α opt for each crystal and the corresponding SPDCsettings are shown in table I.The physical intuition behind the curves in Fig. 3a)is the following. In case of weak to no filtering (top leftin Fig. 3a)), the transmission efficiency is the highestand the spectral purity the lowest. In this weak filteringregime the crystal length and pump bandwidth are suchthat the JSA is as factorizable as it can be without fil-tering. This can also be seen in Fig 4. Examples of suchJSAs can be found in the appendix. Filter bandwidth [nm] S pe c t r a l P u r i t y Unfiltered Purity
GaussianRectangular
Filter Type
FIG. 4. The spectral purities of a ppKTP source with a sincphase-matching function. The solid lines describe the spectralpurity of the resulting photons before filtering. The dashedlines correspond to the purity after passing through the spec-tral filter.
If we now increase filtering, we arrive at the regime ofmoderate filtering, at the center of Fig. 3a). While in-creasing the filtering, the optimal crystal length increasesand the optimal pump bandwidth decreases. This resultsin a relative increase of the transmission efficiency, sincethe unfiltered JSA is now smaller and ’fits easier’ in thefilter bandwidth. The filter also smoothens out the JSAside lobes into a two-dimensional approximate Gaussian.This is the regime with the optimal value for α .In the case of stronger filtering, the losses start to dom-inate. The optimal strategy in this regime is to make theJSA as small as possible, such that as much of the pho-tons can get through. By doing so, the ’intrinsic’ purity,i.e., before filtering, reduces since this configuration doesno longer result in a factorizable state. However, thisreduction of purity is compensated by the spectral fil-ter. This is shown in Fig. 4, where the ’intrinsic’ puritydecreases, but the filtered purity increases.Furthermore this physical picture also explains the dif-ferences between a rectangular and Gaussian filter win-dow. The first difference is that a Gaussian filter allowsfor higher values of α and thus for more photons in aboson-sampling experiment. The second difference is theoptimal filter bandwidth. Both differences can be ex-plained by noting that a rectangular filter window ide-ally only filters out the side lobes. As a result it cannotincrease the factorability of the ’main’ JSA, i.e., the partwithout the side lobes.The results of the Gaussian apodized source cannotbe understood using the aforementioned physical intu-ition. The filter does not improve the spectral puritysince there are no side lobes and the pump bandwidthand crystal length can be chosen such that the JSA isfactorizable. The limiting factor here is group-velocity TABLE I. The values of α opt and the loss budget for a k = 50photon boson-sampling experiment for different crystals ata center wavelength λ c . The corresponding SPDC settings(crystal length L , pump bandwidth σ p and filter bandwidth σ f ) are also listed. The mentioned bandwidths are FWHM ofthe fields.Crystal α opt η TB λ c L σ p σ f (nm) (mm) (nm) (nm)KDP 0.9804 0.8923 830 25 2.3 6KDP R. a > a Rectangular filter window dispersion, which is small around 1582 nm [20].
IV. DISCUSSION
It is well known that the spectral purity of symmetri-cally group-velocity-matched SPDC sources is invariantto changes of either the crystal length or pump band-width, as long as the other one is changed accordingly.However, Fig. 4 shows that relation no longer holds whenfiltering is included. In the regime of strong filtering, α is dominated by the losses. Therefore, the SPDC con-figuration which optimizes α inevitably is the one thatminimizes the losses. Hence the spectral purity reduces,but this is compensated by the strong filtering.In an experiment the non-unity transmission efficiencyof a filter at the maximum of the transmission windowwill be an important source of losses. As a consequence,spectral filtering is only useful when the filter’s maximumtransmission is larger than α f /α , where α f denotes thefiltered α and α the unfiltered case. If the filter’s trans-mission is lower, then the gain in α is not worth theadditional losses.We note that the ideal filter bandwidths of Tab. I arelarger than what is reported in [14]. We attribute thisdifference to two points. Firstly, the model of [14] ap-proximates the sinc phase-matching function as a Gaus-sian. This eliminates the side lobes and hence reducesthe losses. As a consequence, smaller filter bandwidthsare optimal. Secondly, the model of [14] focuses on thesymmetrized heralding efficiency where both photons arefiltered.A final point regarding the spectral filters is that theoptimal filter bandwidths for ppKTP sources are ratherlarge ( >
100 nm). Photons with such large bandwidthsare typically unpractical for multi-photon experimentssince the properties of optical components, e.g., the split-ting ratio of a beam splitter, are rarely constant over sucha wavelength range. These additional constraints on opti-cal components may result in a better classical simulationof boson sampling. Hence it could increase the requiredeffort to do a boson-sampling experiment.
V. CONCLUSION
In conclusion, we have numerically optimized SPDCsources for scattershot boson sampling. Using the re-cently found source quality parameter α [17] we have in-vestigated a number of candidates for building the nextgeneration of SPDC sources.From the results of Tab. I we conclude that SPDCsources in principle allow the demonstration of a quan-tum advantage with boson sampling. The most suitablesource for boson sampling is an apKTP crystal. Such asource can have a maximal source quality α opt = 0 .
99 andhas a corresponding transmission budget of 0 . α opt = 0 .
98 is a good alternative.The optimal source quality for BBO is found to be com-parable with ppKTP and less suited for a boson sam-pling experiment. The fact that these asymmetricallygroup-velocity-matched sources perform less than sym-metrically matches sources is consistent with previousfindings.The limited tolerance for additional losses for theGaussian apodized KTP source suggests that bothwaveguide sources and bulk sources without focusing ofthe pump beam are ideal. Such sources have a singlespatial mode and thus do not suffer from an additionalreduction of distinguishability which is inevitable withfocusing [15].This work can be extended to other SPDC sources suchas [44–46], four-wave mixing sources [47] and to Gaus-sian boson sampling [48]. The latter can be realized byincluding the distinguishability between the signal andidler photons.
ACKNOWLEDGMENTS
The Complex Photonic Systems group acknowledgesfunding from the Nederlandse Wetenschaps Organsiatie(NWO) via QuantERA QUOMPLEX (no. 731473),Veni (Photonic Quantum Simulation) and NWA (No.40017607). The Integrated Quantum Optics group ac-knowledges funding from the European Research Coun-cil (ERC) under the European Unions Horizon 2020 re-search and innovation programme (Grant agreement No. 725366, QuPoPCoRN).
Appendix: Optimal SPDC settings
The effect of the filter bandwidth on the optimal SPDCconfiguration (except for apKTP) can be categorized inthree different regimes. These regimes are the weak,moderate and strong filtering regime. An example of theJSA of a ppKTP source in all three regimes can be seenin Fig. 6.The corresponding SPDC configuration parameterscan be seen in Figure 5. This figure shows that in theweak filtering regime, the bounds on the crystal sizes andpump bandwidth can be reached. Once such a bound isreached, the SPDC configuration loses a parameter tooptimize the JSA factorizability with, meaning that thegeneral trend of matching the crystal length and pumpbandwidth cannot continue anymore. This limits the pu-rity. In case of ppKTP, the limiting factor is the crystallength, whereas in case of a BBO source the maximumpump bandwidth is the limiting factor.
Appendix: Numerical stability
We used a local optimization algorithm to find the op-timal SPDC configuration for different filter bandwidths.Each iteration of this algorithm computes the spectralpurity and losses by discretizing the (filtered) JSA. Sucha numerical approach can fail and/or give wrong results.The algorithm can fail because the problem is not con-vex or that it finds unphysical results (such as a negativecrystal length). The algorithm can give wrong results ifthe discretization of the JSA is too coarse.By bounding the parameter space we guarantee thatthe algorithm does not reach unphysical results. Fur-thermore, we note that optimizing over the whole pa-rameter space, i.e., the filter bandwidths, crystal lengthsand pump bandwidths is not a convex problem. Thisproblem is solved by optimizing the crystal and pumpproperties each time for different filter bandwidths.The discretization of the JSA can cause numerical er-rors. Increasing the number of grid points, i.e., increasingthe resolution, decreases this numerical error. Increasingthe resolution results to a convergence of the result. Un-fortunately, it is not directly known how our numericalcalculation converges to a reliable answer. How to a pri-ori estimate the numerical error for a given discretizationis also unclear.In order to show that our calculations have converged,we simply try different discretizations of the JSA. For ev-ery discretization, we calculate the corresponding sourcequality α and observe how it is varies. Figure 7 showsthat the numerical error originating from this discretiza-tion is small in the limit of more than 2000 (2000 perphoton) grid points. This confirms the validity of our cal-culations. Table II provides an overview of all relevantparameters for the stability of the simulation. P u m p band w i d t h [ n m ] C r ys t a l l eng t h [ mm ] KDPKDP R.ppKTPppKTP R.apKTPBBOBBO R.
FIG. 5. The optimal pump bandwidth and crystal length as a function of the filter bandwidth.TABLE II. The simulation parameters for each crystal. The bounds on the crystal lengths and pump bandwidth are given,just as the range of wavelength over which the JSA is computed. The grid points are the number of steps used to discretizethe entire wavelength rangeCrystal Crystal Length Pump bandwidth Wavelength Grid points Sellmeier constantsminimum maximum minimum maximum minimum maximum(mm) (mm) (nm) (nm) (nm) (nm)KTP 0.5 30 0.1 30 1028 2136 2000 [49, 50]BBO 0.5 40 0.1 30 1008 2093 2000 [51]KDP 0.5 25 0.1 10 780 880 1500 [52][1] A. W. Harrow and A. Montanaro, Quantum computa-tional supremacy, Nature , 203 (2017).[2] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C.Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L.Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen,B. Chiaro, R. Collins, W. Courtney, A. Dunsworth,E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina,R. Graff, K. Guerin, S. Habegger, M. P. Harrigan,M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang,T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang,D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh,A. Korotkov, F. Kostritsa, D. Landhuis, M. Lind-mark, E. Lucero, D. Lyakh, S. Mandr`a, J. R. Mc-Clean, M. McEwen, A. Megrant, X. Mi, K. Michielsen,M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill,M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quin-tana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank,K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Tre-vithick, A. Vainsencher, B. Villalonga, T. White, Z. J.Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Marti- nis, Quantum supremacy using a programmable super-conducting processor, Nature , 505 (2019).[3] S. Aaronson and A. Arkhipov, The Computational Com-plexity of Linear Optics, Theory Comput. , 143 (2013).[4] M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove,S. Aaronson, T. C. Ralph, and A. G. White, PhotonicBoson Sampling in a Tunable Circuit, Science , 794(2013).[5] J. B. Spring, B. J. Metcalf, P. C. Humphreys,W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta,N. Thomas-Peter, N. K. Langford, D. Kundys, J. C.Gates, B. J. Smith, P. G. R. Smith, and I. A. Walms-ley, Boson Sampling on a Photonic Chip, Science ,798 (2013).[6] M. Tillmann, B. Daki´c, R. Heilmann, S. Nolte, A. Sza-meit, and P. Walther, Experimental boson sampling, Nat.Photonics , 540 (2013).[7] A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F.Galv˜ao, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mat-aloni, and F. Sciarrino, Integrated multimode interferom- a) b) c) d) e) f) I d l e r W a v e l eng t h [ n m ] Signal Wavelength [nm] I d l e r W a v e l eng t h [ n m ] Signal Wavelength [nm]Signal Wavelength [nm]1200 1600 2000 1200 1600 2000 1200 1600 2000
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