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Quantum experiments and graphs II: Quantuminterference, computation, and state generationXuemei Gua,b,1, Manuel Erharda,c, Anton Zeilingera,c,1, and Mario Krenna,c,1,2

aInstitute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, 1090 Vienna, Austria; bState Key Laboratory for Novel SoftwareTechnology, Nanjing University, 210023 Nanjing City, China; and cVienna Center for Quantum Science & Technology, Faculty of Physics, University of Vienna,1090 Vienna, Austria

Contributed by Anton Zeilinger, November 28, 2018 (sent for review September 13, 2018; reviewed by Austin Lund and Jianwei Wang)

We present an approach to describe state-of-the-art photonicquantum experiments using graph theory. There, the quan-tum states are given by the coherent superpositions of perfectmatchings. The crucial observation is that introducing complexweights in graphs naturally leads to quantum interference. Thisviewpoint immediately leads to many interesting results, someof which we present here. First, we identify an experimen-tal unexplored multiphoton interference phenomenon. Second,we find that computing the results of such experiments is #P-hard, which means it is a classically intractable problem dealingwith the computation of a matrix function Permanent and itsgeneralization Hafnian. Third, we explain how a recent no-goresult applies generally to linear optical quantum experiments,thus revealing important insights into quantum state genera-tion with current photonic technology. Fourth, we show howto describe quantum protocols such as entanglement swappingin a graphical way. The uncovered bridge between quantumexperiments and graph theory offers another perspective on awidely used technology and immediately raises many follow-upquestions.

quantum experiments | graph theory | linear optics | multiphotonquantum interference | quantum entanglement

Photonic quantum experiments prominently use probabilisticphoton sources in combination with linear optics (1). This

allows for the generation of multipartite quantum entanglementsuch as Greenberger–Horne–Zeilinger (GHZ) states (2–5), Wstates (6), Dicke states (7, 8), or high-dimensional states (9, 10);proof-of-principle experiments of special-purpose quantum com-puting (11–18); or applications such as quantum teleportation(19, 20) and entanglement swapping (21, 22).

Here we show that one can describe all of these quantumexperiments∗ with graph theory. To do this, we generalize arecently found link between graphs and a special type of quan-tum experiments with multiple crystals (23)—which were basedon the computer-inspired concept of entanglement by path iden-tity (24, 25). By introducing complex weights in graphs, we cannaturally describe the operations of linear optical elements, suchas phase shifters and beam splitters, which enables us to describequantum interference effects. This technique allows us to findseveral results: (i) We identify a multiphotonic quantum inter-ference effect which is based on generalization of frustratedpair creation in a network of nonlinear crystals. (Frustrated paircreation is an effect where the amplitudes of two pair-creationevents can constructively or destructively interfere.) Althoughthe two-photon special case of this interference effect wasobserved more than 20 years ago (26), the multiphoton general-ization with many crystals has not been investigated theoreticallyor experimentally. (ii) We find these networks of crystals cannotbe calculated efficiently on a classical computer. The experi-mental output distributions require the summation of weights ofperfect matchings (the weight of a perfect matching is the prod-uct of the weights of all containing edges) in a complex weightedgraph (or, alternatively, probabilities proportional to the matrixfunction Permanent and its generalization Hafnian), which is

#P-hard (27, 28) (a #P-hard problem is at least as difficult as anyproblem in the complexity class #P)—and related to the Boson-Sampling problem. (iii) We show that insights from graph theoryidentify restrictions on the possibility of realizing certain classesof entangled states with current photonic technology. (iv) Thegraph-theoretical description of experiments also leads to a pic-torial explanation of quantum protocols such as entanglementswapping. We expect that this will help in designing or intu-itively understanding novel (high-dimensional) quantum proto-cols. The conceptual ideas that have led to this article are shownin Fig. 1.

Connections between graph theory and quantum physics havebeen drawn in earlier complementary works. A well-knownexample is the so-called graph states, which can be used foruniversal quantum computation (31, 32). That approach hasbeen generalized to continuous-variable quantum computation(33), using an interesting connection between Gaussian statesand graphs (34). Graphs have also been used to study collec-tive phases of quantum systems (35) and used to investigatequantum random networks (36, 37). The bridge between graphsand quantum experiments that we present here is quite differ-ent, thus allowing us to explore entirely different questions. The

Significance

Graph theory can be used to model and explain differentphenomena from physics. In this paper, we show that onecan interpret quantum experiments composed of linear opticsand probabilistic sources with graph theory. The importantunderstanding is that complex weights in the graph natu-rally describe the quantum interference. That point of viewleads to several results: We uncover a yet unexplored typeof multiphoton quantum interference. It can be used to solvequestions that are intractable on a classical computer. Also,the connection points toward a general restriction on cre-ating certain classes of quantum states. In general, it givesanother perspective on a mature photonic technology and willbe significant for future designs of such experiments.

Author contributions: X.G. and M.K. designed research; X.G., M.E., and M.K. performedresearch; X.G., M.E., A.Z., and M.K. wrote the paper; and A.Z. and M.K. supervised theresearch.y

Reviewers: A.L., The University of Queensland; and J.W., Peking University.y

The authors declare no conflict of interest.y

Published under the PNAS license.y1 To whom correspondence may be addressed. Email: anton.zeilinger@univie.ac.at,xmgu@smail.nju.edu.cn, or mario.krenn@univie.ac.at.y

2 Present address: Department of Chemistry, University of Toronto, Toronto, ON M5S 3H6,Canada.y

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1815884116/-/DCSupplemental.y

Published online February 15, 2019.∗The experiments mentioned before all consist of probabilistic photon pair sources andlinear optics. This is what we mean by quantum experiments for the rest of this paper.We show that graphs can describe all of such experiments. Additionally, the property ofperfect matchings corresponds to n-fold coincidence detection, which is widely used inquantum optics experiments.

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Fig. 1. A rough sketch of the influences that have led to the current paper.Three seminal papers (26, 29, 30) have influenced entanglement by pathidentity (25), which itself has led to quantum experiments and graphs inref. 23. Here we connect these ideas with the mature field of research thatinvestigates passive linear optics in the quantum regime. The results of themerger are described in the current paper.

correspondence between graph theory and quantum experimentsis listed in Table 1.

Entanglement by Path Identity and GraphsIn this section, we briefly explain the main ideas from entan-glement by path identity in ref. 25 and quantum experimentsand graphs in ref. 23, which form the basis for the rest of thispaper. The concept of entanglement by path identity shows avery general way to experimentally produce multipartite andhigh-dimensional entanglement. This type of experiment can betranslated into graphs (23). As an example, we show an exper-imental setup which creates a 2D GHZ state in polarization(Fig. 2A). The probabilistic photon pair sources (for example,the nonlinear crystals) are set up in such a way that crystals I andII can create horizontally polarized photon pairs, while crystalsIII and IV produce vertically polarized photon pairs. All of thecrystals are excited coherently and the laser pump power is setsuch that two photon pairs are produced simultaneously.†

The final state is obtained under the condition of four-fold coincidences, which means that all four detectors clicksimultaneously.‡ This can happen only if the two photon pairsoriginate either from crystals I and II or from crystals III and IV.There is no other case to fulfill the fourfold coincidence condi-tion. For example, if crystals I and III fire together, there is nophoton in path d , while there are two photons in path a . Thefinal quantum state after postselection can thus be written as|ψ〉= 1√

2( |H ,H ,H ,H 〉abcd + |V ,V ,V ,V 〉abcd), where H and

V stand for horizontal and vertical polarization respectively, andthe subscripts a , b, c, and d represent the photon’s paths.

One can describe such types of quantum experiments usinggraph theory (23). There, each vertex represents a photon path

†The pair creation process of SPDC is entirely probabilistic. That means, the probabil-ity that two pairs are created in one single crystal is as high as that of the creation oftwo pairs in two crystals. That furthermore means that if creating one pair of photonshas a probability of p, then creating two pairs has the probability p2. In the exper-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-tonic GHZ-state experiments (4). However, the laser power can be adjusted such thatthese cases have a sufficiently low probability (of course, with the drawback of lowercount rates). The same arguments hold for all other examples in this paper (as they dofor most other SPDC-based quantum optics experiments).

‡Most multiphotonic entangled quantum states are created under the condition of N-fold coincidence detection (1). It allows for investigation and application of these statesas long as the photon paths are not combined anymore, such as in a subsequent lin-ear optical setup. In that case, one needs to analyze the perfect matchings after theentire setup. Alternatively, one can use a photon number filter based on quantumteleportation in each output of the setup, as introduced in ref. 20.

and each edge stands for a nonlinear crystal which can prob-abilistically produce a correlated photon pair. Therefore, theexperiment can be described with a graph of four vertices andfour edges depicted in Fig. 2B. A fourfold coincidence is givenby a perfect matching of the graph, which is a subset of edgesthat contains every vertex exactly once. For example, there aretwo subsets of edges (Eab , Edc) and (Eac , Ebd ) in Fig. 2B, whichform the two perfect matchings. Thus, the final quantum stateafter postselection can be seen as the coherent superposition ofall perfect matchings of the graph.

Complex Weighted Graphs—Quantum ExperimentsQuantum Interference. Now we start generalizing the connectionbetween quantum experiments and graphs. The crucial obser-vation is that one can deal with a phase shifter in the quantumexperiment as a complex weight in the graph. When we add phaseshifters in the experiments and all of the crystals produce indistin-guishable photon pairs, the experimental output probability withfourfold postselection is given by the superposition of the perfectmatchings of the graph weighted with a complex number.

As an example shown in Fig. 3A, we insert a phase shifterbetween crystals I and III and all of the four crystals create hor-izontally polarized photon pairs. The phase ϕ is set to a phaseshift of π and the pump power is set such that two photon pairsare created. With the graph–experimental connection, one canalso describe the experimental setup as a graph which is depictedin Fig. 3B. The color of the edge stands for the phase in theexperiments while the width of the edge represents the absolutevalue of the amplitude. To calculate fourfold coincidences fromthe outputs, we need to sum the weights of perfect matchingsof the corresponding graph. There are two perfect matchings ofthe graph, where one is given by crystals III and IV while theother is from crystals I and II. The interference of the two per-fect matchings (which means of the two fourfold possibilities) canbe obtained by varying the relative complex weight e iϕ betweenthem. Therefore, the cancellation of the perfect matchings showsthe destructive interference in the experiment.

More quantitatively, each nonlinear crystal probabilisticallycreates photon pairs from spontaneous parametric down-conversion (SPDC). We follow the theoretical method presentedin refs. 29 and 38 and describe the down-conversion creationprocess as

U ≈ 1 + g(a†b†) +g2

2(a†b†)2 +O(g3), [1]

where a† and b† are single-photon creation operators in pathsa and b, and g is the down-conversion amplitude. The termsof O(g3) and higher are neglected. The quantum state can beexpressed as |ψ〉= U | vac〉, where | vac〉 is the vacuum state.

Table 1. The analogies for quantum experiments and graphtheory

Linear optical quantum experiments Graph theory

Quantum photonic setup including linear Complex weightedoptical elements and nonlinear crystals undirected graphas probabilistic photon pair sources

Optical output path Vertex set SPhotonic modes in optical output path Vertices in vertex set SMode numbers Labels of the verticesPhoton pair correlation EdgesPhase between photonic modes Color of the edgesAmplitude of photonic modes Width of the edgesn-fold coincidence Perfect matching#(terms in quantum state) #(perfect matchings)

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I II

III

IV

III

III IV|V,V> |V,V>

|H,H>|H,H>

A B

Fig. 2. Generation of 2D four-photon GHZ state using entanglement bypath identity (25) and corresponding graph description of the setup (23). (A)An optical setup consists of four probabilistic photon pair sources, for exam-ple nonlinear crystals. The crystals (gray squares) I–IV are pumped coherentlyand the pump power is set in such a way that two photon pairs are pro-duced. Here we take the polarization for simplicity—crystals I and II eachproduce a photon pair with |H, H〉 while crystals III and IV create a pho-ton pair with |V , V〉. The fourfold coincidence requires a photon in eachdetector simultaneously, which can happen only when crystals I and II orcrystals III and IV fire together. (B) The corresponding graph of the exper-iment. Each vertex stands for a photon path and each edge representsone crystal. Thus, the graph has four vertices and four edges. The condi-tion of fourfold coincidence is represented by the perfect matchings of thegraph—a subset of edges that contains every vertex exactly once. Thereare two subsets of edges (Eab, Edc) and (Eac, Ebd) which form the perfectmatchings in the graph. The final output state with postselection is in asuperposition of all of the possibilities. Therefore, it can be seen as a super-position of all of the perfect matchings of the graph, which gives the result|ψ〉= 1√

2( |H, H, H, H〉abcd + |V , V , V , V〉abcd).

Here we neglect the empty modes and higher-order terms andwrite only first-order terms and the fourfold term for second-order spontaneous parametric down-conversion. The full stateup to second order can be seen in SI Appendix. Therefore, thefinal quantum state in our example is

|ψ〉= g( |H ,H 〉ab + |H ,H 〉cd + |H ,H 〉bd + e iϕ |H ,H 〉ac)

+g2(1 + e iϕ) |H ,H ,H ,H 〉abcd + . . ..

[2]

We can see that the fourfold coincidence count rate varieswith the tunable phase ϕ while the twofold coincidence countrate remains constant, which is depicted in Fig. 3C. This isa multiphotonic generalization of two-photon frustrated down-conversion (26) that has never been experimentally observed.

Special-Purpose Quantum Computation. We here show a general-ization of the setup in Fig. 3A, where the experimental resultscannot be calculated efficiently on a classical computer. The out-put requires summation of weights of perfect matchings of acomplex weighted graph, which is a remarkably difficult prob-lem that is #P-hard (27, 28). The experiment consists of Nnonlinear crystals and M optical output paths in total. Wecall this type of experiment “the crystal network” for the restof this paper. One can experimentally adjust the pump powerand phases for every crystal, which allows one to change everysingle weight of the edges of the corresponding graph inde-pendently. The crystals are pumped coherently and the pumppower is set such that n (n <N ) crystals can produce photonpairs and higher-order pair creations can be neglected. Thenwe calculate the 2n-fold coincidence in 2n (2n <M ) outputpaths. Now one could ask, What is the probability of the 2n-fold coincidences in one specific 2n output when all crystals arepumped?

In Fig. 4, we show some examples to answer this question.In the first example, we have in total six output paths (a − f :

M = 6) and nine crystals (N = 9) from which probabilisticallytwo (n = 2) produce photon pairs. Now we calculate the fourfoldprobability for a subset of four output paths (for example, a , b, c,and d highlighted in orange). With the graph–experimental link,a subset of four outputs in the quantum experiment correspondsto a subset of four vertices in the corresponding graph, depictedin orange in Fig. 4A. The experimental outcome corresponds tosumming weights of perfect matchings of the subgraph, whichis related to calculating the Permanent of the submatrix of theadjacency matrix. (An adjacency matrix is a square matrix usedto represent a simple graph. The elements of the matrix standfor the weights of the edges between two vertices.) Therefore, wefind that the probability Pabcd is proportional to the Permanent,Pabcd∝ |Perm(UPs )|2.

For experiments with general arrangements of crystals, the2n-fold probability can be calculated by a generalization of thePermanent—the so-called Hafnian (39), shown in Fig. 4B. Whenthe crystal network consists of a large number of crystals, it isunknown how to efficiently approximate the Hafnian (40, 41). Tothe best of our knowledge, the fastest algorithm to compute theHafnian of an n ×n complex matrix runs inO(n32n/2) time (42).

The task described above is connected to BosonSampling(11, 13–17), which requires the matrix function Permanent tocalculate the experimental results. However, the experimentalimplementation is fundamentally different. In BosonSamplingexperiments to date, single photons undergo a multiphotonicHong–Ou–Mandel effect (43–45) in a passive linear opticalnetwork. In contrast to that, our concept is based solely onprobabilistic pair sources where frustrated pair creation occurs.Computing Hafnians has only recently been investigated bya complementary approach called Gaussian BosonSampling(46–48).

An interesting question is the scaling of expected count ratesof BosonSampling and the approach presented here. In the orig-inal BosonSampling proposal, n pairs of heralded single photons

A B

C

Fig. 3. Interference of perfect matchings. (A) A setup with all crystals pro-ducing horizontally polarized photon pairs. A phase shifter with a phaseof ϕ is inserted between crystals I and III. (B) The corresponding graph ofthe experimental setup. The complex weight eiϕ introduced by the phaseshifter (eiϕ, here ϕ=π) is depicted with different colors. Here red and blueof each edge stand for 0 and π phase shifts. There are two perfect match-ings of the graph, which come from crystals I and II and crystals III and IV,respectively. When one calculates the sum of the perfect matchings, thequantum state is given by |ψ〉= (1 + eiπ) |H, H, H, H〉abcd = 0. This meansthe two perfect matchings cancel each other. (C) When the phase ϕ changesfrom 0 to 2π, one can see the fourfold coincidence [depicted as #(abcd)]count rate changes while the twofold coincidence [for example, numberof photon pairs in outputs a and b, depicted as #(ab)] count rate remainsconstant.

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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 indistinguishablephoton 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 changethe phases and transition amplitudes (the values are shown in SI Appendix). (A, Bottom) The corresponding graph GP and its adjacency matrix adj(GP) forthe 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 fouroutputs relates to summing the weights of perfect matchings of the subgraph with related vertices, which corresponds to computing the matrix functionPermanent of submatrix UPs highlighted in orange. Thus, the probability that a certain arrangement of detectors clicks is proportional to |Perm(UPs )|2. All ofthe 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 coincidencein specific outputs a, b, c, and e. This corresponds to computing the Hafnian of submatrix UHs , which is a generalization of the Permanent. The probabilityPabce is given by the matrix function Hafnian, Pabce∝ |Haf(UHs )|2.

from n SPDC crystals (with emission probability p) are theinput into a linear optical network with m input and outputpaths. The count rate for n-fold coincidences R is RBS ≈ pn .Later, two independent groups discovered a method to exponen-tially increase the count rate, called Gaussian BosonSamplingand Scattershot BosonSampling (46, 49). There, each of them inputs of the BosonSampling device is fed with one out-put of an SPDC crystal (the second SPDC photon is heralded).That means, there are m SPDC crystals (m >n). That leads toan exponential increased count rate for n-fold coincidences ofRSS ≈

(mn

)pn (1− p)m−n .

Estimating the count rates in our approach needs a slightlymore subtle consideration, as our photons are not the input to aBosonSampling device, but their generation itself is in a super-position. Let us look at the example given in Fig. 4A. Here wecompare a complete bipartite graph to Scattershot BosonSam-pling. For a complete bipartite graph, we have two sets of paths{a, c, e} and {b, d , f }. To calculate the probability of detecting afourfold coincidence, we first derive all possible crystal combina-tions that could lead to a fourfold detection. There are

(32

)ways

to choose two elements from the two sets of paths. Therefore,there exist

(32

)×(32

)combinations of crystal pairs that produce

fourfold coincidences. In general, for m2 crystals and 2n-foldcoincidences we have

(mn

)2. Furthermore, each combination canarise due to two (in general n!) indistinguishable crystal com-binations. For example, an (abcd) fourfold detection can arisefrom a photon pair emission from either crystals I and IV orII and VI, as depicted in Fig. 4A. Of course, the relative phasebetween these possibilities determines whether we expect con-structive or destructive interference. The latter case would notcontribute any counts. Since for BosonSampling the phases arerandomly distributed, we average over a uniform phase distribu-tion to account for all possible phase settings. This is equivalentto a 2D random walk. Thus, in general the average magnitudeof the amplitude gives

√n!. Therefore, the count rate is mag-

nified by n!. Finally, the estimated count rate for our approachbased on path identity is RPI ≈

(mn

)2n!pn (1− p)m

2−n . Theratio of the path identity sampling and Scattershot Boson-Sampling thus is RPI

RSS=(mn

)n! (1− p)m(m−1). This exponential

increase is due to the additional number of crystals (whileScattershot BosonSampling uses m crystals, we use m2) andthe coherent superposition of n! possibilities to receive theoutput. We compare now this ratio for two recent experimen-tal implementations of Scattershot BosonSampling. In 2015,a group performed Scattershot BosonSampling with m = 13and n = 3 (50). With P ≈ 0.01, our approach would lead toroughly 350 times more 2n-fold count rates. In 2018, a differ-ent group performed Scattershot BosonSampling with m = 12and up to n = 5 (51). With the same number of modes andphoton pairs, we would expect roughly 25,000 more 2n-foldcount rates. In SI Appendix, we explain the scaling based onan example.

For realistic experimental situations, one needs to care-fully consider the influence of multipair emissions, stimulatedemission, loss of photons (including detection efficiencies),and amount of photon-pair distinguishabilities in connectionwith statements of computation complexity (such as done, forinstance, in refs. 52–55). A full investigation of these veryinteresting questions is beyond the scope of the current paper.

Linear Optics and Graphs. With the complex weights, one canapply the graph method to describe linear optical elements ingeneral linear optical experiments. First, we describe the actionof a beam splitter (BS) with our graph language. A crystal pro-duces one photon pair in paths a and b while no photon is inpath c, as shown in Fig. 5. Therefore, there is an edge betweenvertices a and b and there is no edge connecting vertex c. Theincoming photon from path b propagates to the BS, which givestwo possibilities: reflection to path b or transmission to path c.In the case of reflection, photons in path b stay in path b with an

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Fig. 5. The action of a beam splitter described with a graph. Here we showa simple linear optical setup with one 50:50 beam splitter. Using graph tech-nique, one can describe the setup as a graph depicted at Right. Step 1: Acrystal produces a correlated photon pair in paths a and b and no photongoes to path c. Therefore, there is an edge between vertices a and b andthere is no edge connecting vertex c. Step 2: The photon in path b propa-gates to the beam splitter which will transmit to path c or reflect to path bwith an additional phase of π/2. Therefore, in the case of transmission, theexistent red edge Eab will connect the vertices a and c. While in the case ofreflection, the existent edge Eab gets a complex weight with phase of π/2shown in green.

additional relative phase of π/2. Thus, the correlation betweenpaths a and b will stay and get a relative phase of π/2. This canbe represented as the original red edge keeps connecting ver-tices a and b while the color of the edge changes to green, whichstands for a relative phase shift π/2. In the case of transmission,photons in path b go to path c, which changes the original cor-relation between paths a and b to that between paths a and c.Therefore, the original red edge is changed to connect vertices aand c.

From the description of the BS above, we can derive the fol-lowing general rules for BSs, which we call BS operation: (i) ABS has two input paths v and w , which correspond to vertices vand w of the graph. Take one input path v as the start. (ii) Fortransmission, duplicate the existent edges to connect the adjacentvertices of v with vertex w which stands for the other input pathof the BS. (iii) For reflection, change the colors of the existentedges to the colors which represent a relative phase shift π/2.(iv) Apply steps ii and iii for path w .

Another important optical device in photonic quantum exper-iments is the mode shifter, e.g., half-wave plates for polarizationor holograms for orbital angular momentum (OAM). The actionof mode shifters can also be described within the graph language(Fig. 6A). The crystal produces an orthogonally or horizontallypolarized photon pair in paths a and b. A mode shifter (suchas half-wave plates) is inserted in path a , which will change thephoton’s horizontal polarization to vertical polarization and viceversa in path a . In the graph, we introduce labels for each vertex(small light-gray disks), which indicate the mode numbers of aphoton. For example, vertices a and b carry the labels H and V,which stand for the horizontal and vertical polarizations. All ofthe mode numbers of one photon in one path are included in alarge black circle—a vertex set. In the graph language, the oper-ation of a mode shifter can be represented by changing the labelsof the vertex.

As another example for the use of the graph technique, wedescribe the manipulation of the polarizing beam splitter (PBS)shown in Fig. 6B. In quantum experiments, a PBS transmitshorizontally polarized photons and reflects vertically polarizedphotons with an additional phase of π/2. If the crystal produceshorizontally polarized photon pairs ( |H ,H 〉ab), photons in patha go to path b and photons in path b go to path a . The connectionbetween paths a and b remains. Therefore, the edge betweenvertices a and b stays as the original red one. If the crystal pro-duces orthogonally polarized photon pairs ( |H ,V 〉ab), there are

two photons in path b—one photon comes from path a andanother photon with an additional phase of π/2 comes from pathb because of reflection. Thus, in the corresponding graph, thereare two labeled vertices in vertex set b and there is no vertex invertex set a.

Introducing linear optical elements in the graph representa-tion of quantum experiments allows us to describe a prominentquantum effect—Hong–Ou–Mandel (HOM) interference (57),which is shown in Fig. 6C. HOM interference can be observed iftwo indistinguishable photons propagate to different input pathsof a BS.

By using the BS operation, one can obtain the final graph.When the crystal produces a horizontally polarized photon pair,we can immediately see that the edges between vertex sets aand b vanish. Thus, the experimental setup shows the destruc-tive interference. If the created photons are in orthogonal

A

B

C

Fig. 6. (A) An example for describing mode shifters with a graph. (A, Left) Acrystal generates a polarized photon pair in paths a and b. A half-wave plate(HWP@45) changes the photon’s polarization such that horizontal polariza-tion changes to vertical polarization and vice versa. A, Right depicts thecorresponding graphs. The vertices with labels H or V represent horizon-tal or vertical polarization of photons. Therefore, the label H changes toV in the vertex set a. (B) Graph description for the polarizing beam split-ter (PBS). A PBS can transmit a horizontally polarized photon and reflect avertically polarized photon. When the crystal creates a horizontally polar-ized photon pair, photons in paths a and b transmit to paths b and a. Whenthe crystal produces an orthogonally polarized photon pair, the photon inpath a is transmitted and the photon in path b is reflected with phase ofπ/2. Therefore, there are two correlated photons in path b. In the graph(B, Right), there are two labeled vertices in vertex sets b with a green edgeconnecting them. (C) An optical setup for Hong–Ou–Mandel (HOM) inter-ference. A crystal produces a photon pair in paths a and b which propagateto a 50:50 BS. By using the BS operation, we get the final graph. Now letus look at two cases where the photons are indistinguishable or distin-guishable. For simplicity, we show the example with polarization. When thetwo photons are indistinguishable—all of their mode numbers are identi-cal such as |H, H〉ab—the edges that connect vertices a and b cancel. Theremaining green edges with two vertices in vertex sets a or two vertices invertex sets b show that there are two photons in path a or path b. Thisis a manifestation of the HOM interference. While in the case that theinput photons have orthogonal polarization such as |H, V〉ab, we clearly seethat no interference can be observed. Therefore, the four possible outputsremain ( |H, V〉aa; |H, V〉ab; |V , H〉ab; |V , H〉bb).

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polarization, the superposition of the perfect matchings isnot zero and then no interference can be observed in theexperiment.

Every other linear optical element can be described withgraphs. That is because linear optics do not change the numberof photons and cannot destroy photon pair correlations. Theycan change phases (which changes the complex weight of edges),intrinsic mode numbers (such as polarization or OAM, whichchanges the mode number in the vertex set), or the extrinsicmode number (i.e., the path of the photon, which leads to recon-nection of edges). All of these actions can be described withinour graph method. Thus, every linear optical setup with proba-bilistic photon pair sources corresponds to an undirected graphwith complex weights.

Therefore, we are equipped with the powerful technique ofthe mathematical field of graph theory, which we can now applyto many state-of-the-art photonic experiments.

Restriction for GHZ State Generation. In ref. 23, we have shown arestriction on the generation of high-dimensional GHZ states.The limitation stems from the fact that certain graphs with spe-cial properties (concerning their perfect matchings) cannot exist.Since we have extended the use of graphs to linear optics, thisrestriction applies more generally. We show this restriction byinvestigating a particular linear optical experiment.

To understand this example, let us first analyze the creationof the 2D GHZ state. For creating a three-particle GHZ state,we can connect two crystals with a PBS. If the two crystals bothcreate a Bell state, a three-photonic GHZ state with a trigger in

a is created (shown in Fig. 7A) (58). Extending this to a four-particle GHZ state, we add another crystal that is connectedvia a PBS as depicted in Fig. 7B. (A four-particle polarizationGHZ state can also be created in a simpler way by connectingtwo crystals via a PBS without a trigger with the same setupin Fig. 7A. However, thereby we emphasize the analogy to the3D case.)

Now we are trying exactly the same in a 3D system. Tocreate a 3D GHZ state, we can use two crystals (each gen-erating a 3D entangled photon pair) and connect them witha 3D multiport (10), as shown in Fig. 7C. The graphicaldescription for the setup is depicted in Fig. 7E. There arefive perfect matchings of the final graph. When we calculatethe sum of the perfect matchings (two of them cancel), wecan get the final quantum state written as |ψ〉= 1√

3( | 3, 1, 1〉−

| 2, 0, 0〉− |−1,−1,−1〉)bcd, which describes a 3D three-particleGHZ state. [Three-particle GHZ state can be written as |ψ〉=1√3( | x , y , z 〉+ | x , y , z 〉+ | ¯x , ¯y , ¯z 〉), where m⊥m⊥ ¯m with m =

x , y , z . The properties of entanglement cannot be changed bylocal transformations.]

In exact analogy to the 2D case, we add another crystal tothe setup and connect it with another multiport (Fig. 7D). As inthe 2D case, we would naturally expect to create a four-particleGHZ state in 3D with this setup. However, in this setup, six pho-tons are used (two triggers and four photons for the GHZ state),and therefore the corresponding graph has six vertices. From ref.23 we know that such graphs cannot generate high-dimensionalGHZ states because additional terms (so-called Maverick terms)

A B E

FDC

Fig. 7. Generating high-dimensional multiphotonic states with linear optical setups. (A) An optical setup for creating a 2D three-photon GHZ state. In thisexample, each crystal produces an entangled state |ψ+

⟩= 1/√

2( |H, H〉+ |V , V〉). The photons propagate to a polarizing beam splitter (PBS), and fourfoldcoincidences 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, onecan add more crystals and connect them via many PBSs. (C) In an analogous way, a 3D three-photon GHZ state ( |ψ〉= 1/

√3( | 0, 0〉+ | −1, 1〉+ | 1,−1〉)) 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 aon |+〉= | T〉= 1/

√2( | 0〉+ | −1〉). (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 (detailsin SI Appendix). The quantum state conditioned on fourfold coincidences is obtained by calculating the perfect matchings of the graph. There are fiveperfect matchings and two of them cancel each other, which results in a 3D GHZ state after triggering the photon in path a on | T〉= 1/

√2( | 0〉+ | −1〉).

(F) These graphs describe the experimental setup in D. As expected, it has four perfect matchings (the other four perfect matchings are canceled), threecorresponding to the GHZ state while the fourth one (highlighted in light blue) is the so-called Maverick term.

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occur in the final state. (If the quantum state is independent ofthe trigger photons, then it consists of only four vertices, andthese can be in a 3D GHZ state. Independent means that edgesbetween the trigger vertices and the state vertices do not appearin any perfect matching.) And indeed, when we compute theperfect matchings of the graph, the final quantum state withpostselection is given by |ψ〉= 1

2( | −1,−1, 1, 3〉− | 2, 0, 0, 2〉+

| 3, 1,−1,−1〉+ | −1, 0, 0,−1〉)bcde, which is not a GHZ statebecause of the additional term | −1, 0, 0,−1〉bcde. This is theadditional perfect matching that leads to the Maverick term(Fig. 7F), which comes from the tripled photon pairs emissionof the middle crystal.

For higher dimensions, even more additional terms willappear—which can be understood by perfect matchings ofgraphs. The Maverick term is therefore a genuine manifestationof the graph description in a linear optical quantum experimentwith a probabilistic photon source. Therefore, 2D n-particleGHZ states can be created while the 3D GHZ state with fourparticles is the highest-dimensional entangled GHZ state pro-ducible with linear optics and probabilistic photon sources inthis way (for instance, without exploiting further ancillaryphotons).

Graphical Description for Quantum Protocols. Finally, we showthat using graphs can also help to interpret quantum proto-cols. In Fig. 8, the entanglement swapping is described withgraphs (1, 59). One crystal produces an entangled state |ψ−

⟩=

1√2( | 0, 1〉− | 1, 0〉). Therefore, the initial graph has two edges

between the vertex set a and b and two edges between c andd . The weights of the two edges have a phase difference of π.With the BS operation, we can obtain the final graph. In theend, we obtain all perfect matchings and redraw the graph, whichshows the entanglement swapping. The link between graphsand quantum experiments offers a graphical way to under-stand experimental quantum applications such as entanglementswapping.

ConclusionWe have presented a connection between linear optical quantumexperiments with probabilistic photon pair sources and graphtheory. The final quantum state after postselection emergesas a superposition of graphs (more precisely, as a superpo-sition of perfect matchings). With complex weights in thegraphs, we find interference of perfect matchings which describesthe interference of quantum states. Equipped with that tech-nique, we identify a multiphotonic interference effect and showthat calculating the outcome of such an experiment on aclassical computer is remarkably difficult. Different from theinterference which occurs in the BosonSampling experimentswith linear optics, the underlying effect in our crystal net-work is multiphotonic frustrated photon generation. It wouldbe exciting to see an actual implementation in a laboratory—potentially in integrated platforms which allow for on-chipphoton pair generation (60–67). While we have shown thatthe expected n-fold coincidence counts will be larger thanin conventional BosonSampling systems, an important ques-tion is how these systems compete under realistic experimentalsituations.

Another important question is how these setups can be appliedto tasks in quantum chemistry, such as calculations of vibrationalspectra of molecules (68, 69), or topological indexes of molecules(70), or graph theory problems (71).

So far, we have focused on n-fold coincidences with onephoton per path, which is directly connected to perfect match-ings. A generalized graph description which allows for arbitraryphotons per path would also be a very interesting questionfor future research, which will need to exploit not only per-

A

B

Fig. 8. Experimental diagram for entanglement swapping and correspond-ing graph description. (A) An experimental setup for entanglement swap-ping. Each crystal probabilistically generates an entangled state |ψ−

⟩=

1√2

( | 0, 1〉− | 1, 0〉). When the photons emerge in paths b and c after the

BS, the two-photon state in a and d is projected into the Bell state in |ψ−⟩.

(B) Here we show the experiment using graphs. The initial |ψ−⟩

states(depicted in dashed box I) both have a relative phase of π, which is repre-sented by edges with different colors (red and blue). Using the BS operation,we get the final graph shown at Right. There are eight perfect matchings,and four of them cancel (highlighted in gray). Due to the symmetry in thequantum state (for example, | 0, 0, 1, 1〉abcd = | 0, 1, 0, 1〉acbd), we rearrangethe edges between different vertices after identifying perfect matchings.

With ei3π

2 ei3π

2 = eiπei2π = eiπe0, perfect matching of two purple edges canbe redescribed as one edge in red and another in blue. The perfect match-ing for green edges is depicted in the same way. Finally, we obtain the finalgraph shown in dashed box II. From the two dashed boxes, we can clearlysee the swapping of quantum entanglement.

fect matchings, but also more general techniques in matchingtheory.

With this connection, we uncovered restrictions on classesof quantum states that can be created using state-of-the-art photonic experiments with probabilistic photon sources,in particular, higher-dimensional GHZ states. The graph–experimental link could be used for investigating restrictionsof other, much larger types of quantum states (72, 73) orcould help in understanding the (non)constructability of cer-tain 2D states. Restrictions for the generation of quantum stateshave been found before, using properties of Fock modes (74)for instance, and it would be interesting to discover whetherthose two independent techniques could be merged. Also severerestrictions on high-dimensional Bell-state measurements areknown (75), which limits the application of protocols such ashigh-dimensional teleportation. The application of the graph-theory link to such types of quantum measurements would beworthwhile.

As an example, we have shown that entanglement swap-ping can be understood with graphs. A different graphicalrepresentation has been developed to describe quantum pro-cesses at a more abstract level (76, 77). Furthermore, directedgraphs have recently been investigated to simplify certain cal-culations in quantum optics, by representing creation and anni-hilation operators in a visual way (78–80). A combination ofthese pictorial approaches with our methods could hopefully

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improve the abstraction and intuitive understanding of quantumprocesses.

In ref. 23, we have shown that every experiment (based oncrystal configurations as shown in Fig. 2) corresponds to an undi-rected graph and vice versa. It is still an open question whetherfor every undirected weighted graph, one can find a linear opticalsetup without path identification. This is an important questionfor the design of new experiments.

Our method can conveniently describe linear optical exper-iments with probabilistic photon sources. It will be useful tounderstand how the formalism can be extended to other typesof probabilistic sources, such as single-photon sources based onweak lasers (81), or three-photon sources based on cascadeddown-conversion (82, 83), or general multiphotonic sources (84).Can it also be applied to other (nonphotonic) quantum systemswith a probabilistic source of quanta?

A final, very important question is how to escape the restric-tions imposed by the graph-theory link. Deterministic quan-tum sources (85–87) would need an adaption of the descrip-tion, and it is not yet known how to describe active feedfor-ward (88–90). Can they be described with graphs? What arethe techniques that cannot be described in the way presentedhere?

ACKNOWLEDGMENTS. The authors thank Armin Hochrainer, JohannesHandsteiner, and Kahan Dare for useful discussions and valuable commentson the manuscript. X.G. thanks Lijun Chen for support. This work was sup-ported by the Austrian Academy of Sciences and by the Austrian ScienceFund with Spezialforschungsbereiche F40 (Foundations and Applicationsof Quantum Science). X.G. acknowledges support from the National Nat-ural Science Foundation of China (Grant 61771236) and its Major Program(Grants 11690030 and 11690032), the National Key Research and Develop-ment Program of China (Grant 2017YFA0303700), and a scholarship fromthe China Scholarship Council.

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