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(high-dimensional) this novel entanglement understanding that as itively expect such protocols We pic- quantum swapping. a The of to (iv) explanation leads technology. also torial experiments photonic of current description classes with graph-theoretical certain states realizing entangled of possibility of theory the graph on from insights restrictions Boson- that identify the show to We (iii) related any problem. #P)—and as Sampling class difficult complexity as least the at in is problem problem #P-hard (a 28) (27, #P-hard 2 1 the under Published interest.y of conflict no University. declare y Peking authors J.W., The and Queensland; of University The A.L., Reviewers: 1073/pnas.1815884116/-/DCSupplemental. y at online information supporting contains article This the supervised M.K. and A.Z. and paper; the wrote performed M.K. M.K. and and research. M.E., A.Z., X.G., M.E., research; X.G., designed research; M.K. and X.G. contributions: Author ∗ rsn drs:Dprmn fCeity nvriyo oot,Trno NMS3H6, M5S ON Toronto, Toronto, Canada. of [email protected], University Chemistry, of Email: Department address: Present addressed. be [email protected]. y may or [email protected], correspondence whom To eso htgah a eciealo uheprmns diinly h rpryof property the Additionally, experiments. such of all describe can graphs that show We unu pisexperiments. optics quantum efc acig orsod to corresponds matchings perfect ierotc.Ti swa ema yqatmeprmnsfrters fti paper. this of rest the for experiments quantum by mean we what is This optics. linear h xeiet etoe eoealcnito rbblsi htnpi ore and sources pair photon probabilistic of consist all before mentioned experiments The esgicn o uuedsgso uhexperiments. such gives of designs it future general, for will cre- and significant In technology be photonic on states. mature a restriction quantum on perspective of general another classes Also, a computer. certain toward classical ating points a connection on solve type intractable the to unexplored used are yet be that can a questions It uncover interference. view quantum We of multiphoton natu- results: of point graph several That the to interference. quantum leads in the weights important describe complex The rally theory. that graph is one with understanding that different sources show optics probabilistic explain linear we of and and paper, composed experiments model this quantum interpret In to can physics. used from be phenomena can theory Graph Significance oncin ewe rp hoyadqatmpyishave physics quantum and theory graph between Connections y y a,c,1,2 NSlicense.y PNAS odcicdnedtcin hc swdl sdin used widely is which detection, coincidence n-fold b tt e aoaoyfrNvlSoftware Novel for Laboratory Key State www.pnas.org/lookup/suppl/doi:10. 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PHYSICS and each edge stands for a nonlinear crystal which can prob- abilistically produce a correlated photon pair. Therefore, the experiment can be described with a graph of four vertices and four edges depicted in Fig. 2B. A fourfold coincidence is given by a perfect matching of the graph, which is a subset of edges that contains every vertex exactly once. For example, there are two subsets of edges (Eab , Edc ) and (Eac , Ebd ) in Fig. 2B, which form the two perfect matchings. Thus, the final quantum state after postselection can be seen as the coherent superposition of all perfect matchings of the graph. Complex Weighted Graphs—Quantum Experiments Quantum Interference. Now we start generalizing the connection Fig. 1. A rough sketch of the influences that have led to the current paper. between quantum experiments and graphs. The crucial obser- Three seminal papers (26, 29, 30) have influenced entanglement by path vation is that one can deal with a phase shifter in the quantum identity (25), which itself has led to quantum experiments and graphs in experiment as a complex weight in the graph. When we add phase ref. 23. Here we connect these ideas with the mature field of research that shifters in the experiments and all of the crystals produce indistin- investigates passive linear optics in the quantum regime. The results of the guishable photon pairs, the experimental output probability with merger are described in the current paper. fourfold postselection is given by the superposition of the perfect matchings of the graph weighted with a complex number. As an example shown in Fig. 3A, we insert a phase shifter correspondence between and quantum experiments between crystals I and III and all of the four crystals create hor- is listed in Table 1. izontally polarized photon pairs. The phase ϕ is set to a phase shift of π and the pump power is set such that two photon pairs Entanglement by Path Identity and Graphs are created. With the graph–experimental connection, one can In this section, we briefly explain the main ideas from entan- also describe the experimental setup as a graph which is depicted glement by path identity in ref. 25 and quantum experiments in Fig. 3B. The color of the edge stands for the phase in the and graphs in ref. 23, which form the basis for the rest of this experiments while the width of the edge represents the absolute paper. The concept of entanglement by path identity shows a value of the amplitude. To calculate fourfold coincidences from very general way to experimentally produce multipartite and the outputs, we need to sum the weights of perfect matchings high-dimensional entanglement. This type of experiment can be of the corresponding graph. There are two perfect matchings of translated into graphs (23). As an example, we show an exper- the graph, where one is given by crystals III and IV while the imental setup which creates a 2D GHZ state in polarization other is from crystals I and II. The interference of the two per- (Fig. 2A). The probabilistic photon pair sources (for example, fect matchings (which means of the two fourfold possibilities) can the nonlinear crystals) are set up in such a way that crystals I and be obtained by varying the relative complex weight eiϕ between II can create horizontally polarized photon pairs, while crystals them. Therefore, the cancellation of the perfect matchings shows III and IV produce vertically polarized photon pairs. All of the the destructive interference in the experiment. crystals are excited coherently and the laser pump power is set More quantitatively, each nonlinear crystal probabilistically such that two photon pairs are produced simultaneously.† creates photon pairs from spontaneous parametric down- The final state is obtained under the condition of four- conversion (SPDC). We follow the theoretical method presented fold coincidences, which means that all four detectors click in refs. 29 and 38 and describe the down-conversion creation simultaneously.‡ This can happen only if the two photon pairs process as originate either from crystals I and II or from crystals III and IV. There is no other case to fulfill the fourfold coincidence condi- g 2 Uˆ ≈ 1 + g(ˆa†bˆ†) + (ˆa†bˆ†)2 + O(g 3), [1] tion. For example, if crystals I and III fire together, there is no 2 photon in path d, while there are two photons in path a. The final quantum state after postselection can thus be written as where aˆ† and bˆ† are single-photon creation operators in paths | ψi = √1 ( | H , H , H , H i + | V , V , V , V i ) H 2 abcd abcd , where and a and b, and g is the down-conversion amplitude. The terms V stand for horizontal and vertical polarization respectively, and of O(g 3) and higher are neglected. The quantum state can be the subscripts a, b, c, and d represent the photon’s paths. expressed as | ψi = Uˆ | vaci, where | vaci is the vacuum state. One can describe such types of quantum experiments using graph theory (23). There, each vertex represents a photon path Table 1. The analogies for quantum experiments and graph theory † The pair creation process of SPDC is entirely probabilistic. That means, the probabil- Linear optical quantum experiments Graph theory ity that two pairs are created in one single crystal is as high as that of the creation of two pairs in two crystals. That furthermore means that if creating one pair of photons Quantum photonic setup including linear Complex weighted 2 has a probability of p, then creating two pairs has the probability p . In the exper- optical elements and nonlinear crystals undirected graph iment depicted in Fig. 2A, with some probability, more than two pairs are created. These higher-order photon pairs are the main source of reduced fidelity in multipho- as probabilistic photon pair sources tonic GHZ-state experiments (4). However, the laser power can be adjusted such that Optical output path Vertex set S these cases have a sufficiently low probability (of course, with the drawback of lower Photonic modes in optical output path Vertices in vertex set S count rates). The same arguments hold for all other examples in this paper (as they do Mode numbers Labels of the vertices for most other SPDC-based quantum optics experiments). Photon pair correlation Edges ‡Most multiphotonic entangled quantum states are created under the condition of N- fold coincidence detection (1). It allows for investigation and application of these states Phase between photonic modes Color of the edges as long as the photon paths are not combined anymore, such as in a subsequent lin- Amplitude of photonic modes Width of the edges ear optical setup. In that case, one needs to analyze the perfect matchings after the n-fold coincidence Perfect matching entire setup. Alternatively, one can use a photon number filter based on quantum #(terms in quantum state) #(perfect matchings) teleportation in each output of the setup, as introduced in ref. 20.

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Each crystals fire iment. when IV and only III happen crystals can which simultaneously, detector with pair ton ue al. et Gu is in example our seen in be state can second- quantum state final order for full second term The to fourfold down-conversion. up the parametric and spontaneous terms order first-order only write pro- each are II pairs and photon I two with simplicity—crystals that pair for way photon polarization a a the such produce take in we set Here is duced. power coherently pump pumped are the I–IV exam- squares) and for (gray sources, crystals pair The crystals. photon nonlinear probabilistic (A) ple four (23). of setup consists the setup of optical description An graph corresponding and (25) identity path 2. Fig. ar n ihrodrpi rain a engetd Then neglected. be can the creations calculate pair we inde- pump higher-order the graph that and and corresponding such pairs coherently set the pumped is of every are power change edges crystals to The the one power pendently. allows of pump rest which the weight the crystal, adjust single every for experimentally for network” can phases crystal One and of “the paper. consists experiment this of experiment of type The this 28). call and prob- (27, a crystals difficult #P-hard of remarkably nonlinear is matchings a that perfect is which of lem graph, weights weighted results of complex experimental out- summation The the computer. requires where classical a put 3A, on Fig. efficiently calculated in be cannot setup the of ization is Computation. Quantum This Special-Purpose 3C. observed. experimentally Fig. been never down- in has frustrated that varies depicted two-photon (26) conversion of rate is generalization which count multiphotonic a constant, coincidence remains fourfold phase rate the tunable that the with see can We ntefis xml,w aei oa i uptpts(a paths output six total in have we example, first the In the of probability the is specific pumped? What one in ask, coincidences could fold one Now paths. 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BosonSampling presented inal approach the and BosonSampling of BosonSampling by Gaussian investigated called on been approach solely (46–48). recently based complementary only is a has occurs. creation concept optical Hafnians pair our frustrated linear Computing where that, sources passive to pair a probabilistic contrast multiphotonic in a In undergo (43–45) network. photons effect single Hong–Ou–Mandel BosonSampling date, In to experimental different. to experiments the fundamentally Permanent However, is function results. implementation matrix experimental the the requires calculate which 13–17), (11, the compute to algorithm is fastest an it of the Hafnian crystals, knowledge, our of To of 41). number (40, best Hafnian large the the approximate a efficiently to of how consists unknown network When 4B. crystal Fig. in the shown (39), Hafnian so-called Permanent—the 2n C AB abcd nitrsigqeto stesaigo xetdcutrates count expected of scaling the is question interesting An BosonSampling to connected is above described task The the crystals, of arrangements general with experiments For ϕ fl rbblt a ecluae yagnrlzto fthe of generalization a by calculated be can probability -fold 6 = d h orsodn rp of graph corresponding The (B) III. and I crystals between inserted is |Perm ∝ ihihe noag) ihtegaheprmna link, graph–experimental the With orange). in highlighted n iecytl (N crystals nine and ) eu ihalcytl pro- crystals all with setup A (A) matchings. perfect of Interference 2 = i ϕ here , rdc htnpis o ecluaetefourfold the calculate we Now pairs. photon produce ) n a e h orodcicdne[eitdas [depicted coincidence fourfold the see can one , n (U ϕ × = P s n π )| sdpce ihdfeetclr.Hr e n blue and red Here colors. different with depicted is ) ope arxrn in runs matrix complex 2 . a | P ψ and π i abcd = hs hfs hr r w efc match- perfect two are There shifts. phase eitdas depicted b, (1 9 = spootoa otePermanent, the to proportional is n + ar fhrle igephotons single heralded of pairs e rmwihprobabilistically which from ) i π ) | H e , hntephase the When ) NSLts Articles Latest PNAS H i ϕ ]cutrt remains rate count #(ab)] , H nrdcdb h phase the by introduced O , H (n i abcd 3 2 n = /2 .Ti means This 0. ) ie(42). time ϕ #(abcd changes a | , f9 of 3 b , c )] ,

PHYSICS A B

Fig. 4. Quantum experiments and computation complexity. (A, Top) An experiment consisting of 9 nonlinear crystals (with labels I–IX) and 18 phase shifters (gold-colored lines). They are arranged such that paths a, c, and e are parallel. All of the crystals are pumped coherently and can produce indistinguishable photon pairs. The pump power is set in such a way that two crystals can produce photon pairs. One can adjust the phase shifters and pump power to change the phases and transition amplitudes (the values are shown in SI Appendix). (A, Bottom) The corresponding graph GP and its adj(GP ) for the setup. The ordering of the columns and rows is (a, c, e, b, d, and f). Calculating fourfold coincidences in one specific subset path (a, b, c, and d) of four outputs relates to summing the weights of perfect matchings of the subgraph with related vertices, which corresponds to computing the matrix function 2 Permanent of submatrix UPs highlighted in orange. Thus, the probability that a certain arrangement of detectors clicks is proportional to |Perm(UPs )| . All of the combinations for the fourfold coincidence are depicted in the histogram (details in SI Appendix). (B, Top) A crystal network that shows the general case. The 9 crystals and 18 phase shifters are randomly put in order. Analogous to A, the pump power is also set such that two photon pairs can be created. (B, Bottom) The corresponding graph GH and its adjacency matrix adj(GH), where the ordering is the same as UP . Again, we calculate the fourfold coincidence

in specific outputs a, b, c, and e. This corresponds to computing the Hafnian of submatrix UHs , which is a generalization of the Permanent. The probability 2 Pabce is given by the matrix function Hafnian, Pabce ∝ |Haf(UHs )| .

from n SPDC crystals (with emission probability p) are the nified by n!. Finally, the estimated count rate for our approach input into a linear optical network with m input and output m2 n m2−n n based on path identity is RPI ≈ n n!p (1 − p) . The paths. The count rate for n-fold coincidences R is RBS ≈ p . ratio of the path identity sampling and Scattershot Boson- Later, two independent groups discovered a method to exponen- Sampling thus is RPI = mn! (1 − p)m(m−1). This exponential tially increase the count rate, called Gaussian BosonSampling RSS n increase is due to the additional number of crystals (while and Scattershot BosonSampling (46, 49). There, each of the 2 m inputs of the BosonSampling device is fed with one out- Scattershot BosonSampling uses m crystals, we use m ) and put of an SPDC crystal (the second SPDC photon is heralded). the coherent superposition of n! possibilities to receive the That means, there are m SPDC crystals (m > n). That leads to output. We compare now this ratio for two recent experimen- an exponential increased count rate for n-fold coincidences of tal implementations of Scattershot BosonSampling. In 2015, m n m−n a group performed Scattershot BosonSampling with m = 13 RSS ≈ n p (1 − p) . Estimating the count rates in our approach needs a slightly and n = 3 (50). With P ≈ 0.01, our approach would lead to more subtle consideration, as our photons are not the input to a roughly 350 times more 2n-fold count rates. In 2018, a differ- BosonSampling device, but their generation itself is in a super- ent group performed Scattershot BosonSampling with m = 12 position. Let us look at the example given in Fig. 4A. Here we and up to n = 5 (51). With the same number of modes and compare a complete to Scattershot BosonSam- photon pairs, we would expect roughly 25,000 more 2n-fold pling. For a , we have two sets of paths count rates. In SI Appendix, we explain the scaling based on {a, c, e} and {b, d, f }. To calculate the probability of detecting a an example. fourfold coincidence, we first derive all possible crystal combina- For realistic experimental situations, one needs to care- 3 fully consider the influence of multipair emissions, stimulated tions that could lead to a fourfold detection. There are  ways 2 emission, loss of photons (including detection efficiencies), to choose two elements from the two sets of paths. Therefore, 3 3 and amount of photon-pair distinguishabilities in connection there exist 2 × 2 combinations of crystal pairs that produce 2 with statements of computation complexity (such as done, for fourfold coincidences. In general, for m crystals and 2n-fold instance, in refs. 52–55). A full investigation of these very m2 coincidences we have n . Furthermore, each combination can interesting questions is beyond the scope of the current paper. arise due to two (in general n!) indistinguishable crystal com- binations. For example, an (abcd) fourfold detection can arise Linear Optics and Graphs. With the complex weights, one can from a photon pair emission from either crystals I and IV or apply the graph method to describe linear optical elements in II and VI, as depicted in Fig. 4A. Of course, the relative phase general linear optical experiments. First, we describe the action between these possibilities determines whether we expect con- of a beam splitter (BS) with our graph language. A crystal pro- structive or destructive interference. The latter case would not duces one photon pair in paths a and b while no photon is in contribute any counts. Since for BosonSampling the phases are path c, as shown in Fig. 5. Therefore, there is an edge between randomly distributed, we average over a uniform phase distribu- vertices a and b and there is no edge connecting vertex c. The tion to account for all possible phase settings. This is equivalent incoming photon from path b propagates to the BS, which gives to a 2D random walk. Thus,√ in general the average magnitude two possibilities: reflection to path b or transmission to path c. of the amplitude gives n!. Therefore, the count rate is mag- In the case of reflection, photons in path b stay in path b with an

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PHYSICS polarization, the superposition of the perfect matchings is a is created (shown in Fig. 7A) (58). Extending this to a four- not zero and then no interference can be observed in the particle GHZ state, we add another crystal that is connected experiment. via a PBS as depicted in Fig. 7B. (A four-particle polarization Every other linear optical element can be described with GHZ state can also be created in a simpler way by connecting graphs. That is because linear optics do not change the number two crystals via a PBS without a trigger with the same setup of photons and cannot destroy photon pair correlations. They in Fig. 7A. However, thereby we emphasize the analogy to the can change phases (which changes the complex weight of edges), 3D case.) intrinsic mode numbers (such as polarization or OAM, which Now we are trying exactly the same in a 3D system. To changes the mode number in the vertex set), or the extrinsic create a 3D GHZ state, we can use two crystals (each gen- mode number (i.e., the path of the photon, which leads to recon- erating a 3D entangled photon pair) and connect them with nection of edges). All of these actions can be described within a 3D multiport (10), as shown in Fig. 7C. The graphical our graph method. Thus, every linear optical setup with proba- description for the setup is depicted in Fig. 7E. There are bilistic photon pair sources corresponds to an undirected graph five perfect matchings of the final graph. When we calculate with complex weights. the sum of the perfect matchings (two of them cancel), we Therefore, we are equipped with the powerful technique of can get the final quantum state written as | ψi = √1 ( | 3, 1, 1i− the mathematical field of graph theory, which we can now apply 3 | 2, 0, 0i−|−1, −1, −1i) , which describes a 3D three-particle to many state-of-the-art photonic experiments. bcd GHZ state. [Three-particle GHZ state can be written as | ψi = √1 ( | x, y, zi + | x¯,y ¯,z ¯i + | x¯, y¯, z¯i), where m⊥m¯ ⊥m¯ with m = Restriction for GHZ State Generation. In ref. 23, we have shown a 3 restriction on the generation of high-dimensional GHZ states. x, y, z. The properties of entanglement cannot be changed by The limitation stems from the fact that certain graphs with spe- local transformations.] cial properties (concerning their perfect matchings) cannot exist. In exact analogy to the 2D case, we add another crystal to Since we have extended the use of graphs to linear optics, this the setup and connect it with another multiport (Fig. 7D). As in restriction applies more generally. We show this restriction by the 2D case, we would naturally expect to create a four-particle investigating a particular linear optical experiment. GHZ state in 3D with this setup. However, in this setup, six pho- To understand this example, let us first analyze the creation tons are used (two triggers and four photons for the GHZ state), of the 2D GHZ state. For creating a three-particle GHZ state, and therefore the corresponding graph has six vertices. From ref. we can connect two crystals with a PBS. If the two crystals both 23 we know that such graphs cannot generate high-dimensional create a Bell state, a three-photonic GHZ state with a trigger in GHZ states because additional terms (so-called Maverick terms)

AB E

C D F

Fig. 7. Generating high-dimensional multiphotonic states with√ linear optical setups. (A) An optical setup for creating a 2D three-photon GHZ state. In this example, each crystal produces an entangled state | ψ+ = 1/ 2( | H, Hi + | V, Vi). The photons propagate to a polarizing beam splitter (PBS), and fourfold coincidences lead to a 2D three-photon GHZ state (where the photon in path a acts as a trigger). (B) For generating high–photon-number√ GHZ states, one can add more crystals and connect them via many PBSs. (C) In an analogous way, a 3D three-photon GHZ state ( | ψi = 1/ 3( | 0, 0i + | −1, 1i + | 1, −1i)) has been created recently, by connecting two crystals (each producing a 3D entangled photon pair) with a 3D multiport (MP) (10). The MP consists of a reflection (R, such as mirrors),√ a spiral-phase plate (SPP), a BS, an OAM mode sorter (56), and a coherent mode projection (CMP) which projects the photon in path a on | +i = | Ti = 1/ 2( | 0i + | −1i). (C, Bottom) The corresponding transformation is described in ref. 10. (D) To create a higher-dimensional GHZ state, we now want to extend the setup to create a 3D GHZ state with four particles. However, since this setup uses six photons, we expect (due to the result in ref. 23) to get an additional term in the final quantum state after postselection. (E) The graphs describing the setup in C, where the vertex set (large black circle) shows the mode numbers of the photons. The initial state shows three connections for each vertex set, which stands for the initial 3D entanglement (details in SI Appendix). The quantum state conditioned on fourfold coincidences is obtained by calculating the perfect matchings of the graph.√ There are five perfect matchings and two of them cancel each other, which results in a 3D GHZ state after triggering the photon in path a on | Ti = 1/ 2( | 0i + | −1i). (F) These graphs describe the experimental setup in D. As expected, it has four perfect matchings (the other four perfect matchings are canceled), three corresponding to the GHZ state while the fourth one (highlighted in light blue) is the so-called Maverick term.

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PHYSICS improve the abstraction and intuitive understanding of quantum A final, very important question is how to escape the restric- processes. tions imposed by the graph-theory link. Deterministic quan- In ref. 23, we have shown that every experiment (based on tum sources (85–87) would need an adaption of the descrip- crystal configurations as shown in Fig. 2) corresponds to an undi- tion, and it is not yet known how to describe active feedfor- rected graph and vice versa. It is still an open question whether ward (88–90). Can they be described with graphs? What are for every undirected weighted graph, one can find a linear optical the techniques that cannot be described in the way presented setup without path identification. This is an important question here? for the design of new experiments. Our method can conveniently describe linear optical exper- ACKNOWLEDGMENTS. The authors thank Armin Hochrainer, Johannes Handsteiner, and Kahan Dare for useful discussions and valuable comments iments with probabilistic photon sources. It will be useful to on the manuscript. X.G. thanks Lijun Chen for support. This work was sup- understand how the formalism can be extended to other types ported by the Austrian Academy of Sciences and by the Austrian Science of probabilistic sources, such as single-photon sources based on Fund with Spezialforschungsbereiche F40 (Foundations and Applications weak lasers (81), or three-photon sources based on cascaded of Quantum Science). X.G. acknowledges support from the National Nat- ural Science Foundation of China (Grant 61771236) and its Major Program down-conversion (82, 83), or general multiphotonic sources (84). (Grants 11690030 and 11690032), the National Key Research and Develop- Can it also be applied to other (nonphotonic) quantum systems ment Program of China (Grant 2017YFA0303700), and a scholarship from with a probabilistic source of quanta? the China Scholarship Council.

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