Abstract
Graphs have provided an expressive mathematical tool to model quantummechanical devices and systems. In particular, it has been recently discovered that graph theory can be used to describe and design quantum components, devices, setups and systems, based on the twodimensional lattice of parametric nonlinear optical crystals and linear optical circuits, different to the standard quantum photonic framework. Realizing such graphtheoretical quantum photonic hardware, however, remains extremely challenging experimentally using conventional technologies. Here we demonstrate a graphtheoretical programmable quantum photonic device in verylargescale integrated nanophotonic circuits. The device monolithically integrates about 2,500 components, constructing a synthetic lattice of nonlinear photonpair waveguide sources and linear optical waveguide circuits, and it is fabricated on an eightinch silicononinsulator wafer by complementary metal–oxide–semiconductor processes. We reconfigure the quantum device to realize and process complexweighted graphs with different topologies and to implement different tasks associated with the perfect matching property of graphs. As two nontrivial examples, we show the generation of genuine multipartite multidimensional quantum entanglement with different entanglement structures, and the measurement of probability distributions proportional to the modulussquared hafnian (permanent) of the graph’s adjacency matrices. This work realizes a prototype of graphtheoretical quantum photonic devices manufactured by verylargescale integration technologies, featuring arbitrary programmability, high architectural modularity and massive manufacturing scalability.
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Main
Graph theory that can be used to model the pairwise relation between objects provides a powerful tool to pictorially describe quantum devices and systems. For example, graph states are the key resource for measurementbased universal quantum computing^{1,2}. Quantum walks on graphs allow the simulations of transport processes in networks^{3,4,5}. The use of graphs allows the characterizations of quantum correlations^{6} and investigations of quantum networks^{7}. Recently, different to these graph quantum connections, another interesting modelanddevice correspondence between the abstract graph theory and the quantum photonic experiment has been proposed^{8,9,10,11}. In this framework, graphs can virtually describe different quantum photonic components, devices, setups and systems, based on nonlinear photonpair sources and linear optical circuits. Graph theory can be adopted to interpret, model and design diverse advanced quantum photonic experiments^{8,9}. Adopting the visualizability and mathematical machinery of graphs, it allows the discovery of complex entanglement resources and even previously unexplored capabilities for quantum technologies^{10,11}. However, realizing graphbased quantum devices remains significantly challenging experimentally, as it requires global quantum coherence over the device to impose genuine multiprocess quantum interference.
In this work, we demonstrate a graphtheoretical quantum photonic device of nonlinear optical sources and linear optical circuits by verylargescale integration (VLSI) of silicon quantum photonics. The topologies of graphs are physically defined by the connectivity of nonlinear optical sources and by the pathway of single photons in linear optical circuits, and can be arbitrarily reprogrammed by reconfiguring the device structures. The device is able to implement very general linear optical quantum experiments. As two examples, we reprogram it to generate and verify complex structures of genuine multipartite multidimensional entanglement, as well as measure the distributions of perfect matchings of general (bipartite) graphs corresponding to the modulussquared hafnian (permanent) matrix functions. Note that the graphtheoretical quantum devices here are different to the graph states for oneway quantum computing^{1,2} or quantum walk on graphs^{3,4,5} (Supplementary Section 7).
Figure 1a–c illustrates the correspondence between graph theory and quantum device, describing pairwise relations in mathematical and physical representations, respectively. The pairwise relation can be described by a complexweighted undirected graph G = (V, E), with a set of edges E that connects pairs of vertices V, with n vertices in total. Determining the number of perfect matchings of a graph (a perfect matching is a subgraph in which every vertex is linked to exactly one edge) is equivalent to estimating the hafnian function of the adjacency matrix of the graph, that is, #P hard^{12}. The original scheme was proposed in bulk optics, to map an abstract graph to a physical setup^{8,9,10,11}. In such a mapping, each pathway of a single photon represents a vertex, and each nonlinear crystal represents an edge. The detected multiphoton quantum correlations at the output of the device returns the number of perfect matchings of the graphs. This correspondence promises a type of versatile quantum photonic device based on graph theory, and could provide a fresh perspective on the existing ones^{13,14,15}.
Retaining quantum coherence over the entire device is the key^{16,17}. We achieve this by ensuring all the processes contributing to multiphoton correlations are quantummechanically indistinguishable. Figure 1a shows an example of the bulk optical scheme for implementing the quantum device. Pairs of single photons, generated in different crystals and routed along different pathways to the same detectors, are no longer distinguishable and undergo quantum interference of identical processes. At each crystal, pump photons must synchronously meet with the incoming single photons from the previous crystal. Pump beams must simultaneously reach the crystals positioned in the same column. That is, one cannot identify the whichsource information of singlephoton generation. In addition, pumps and single photons have different colours and they typically propagate noncollinearly in bulk optics. Thus, retaining global coherence of the device requires complex and precise control of manyphoton wavefunctions in the temporal, spatial and spectral domains. Moreover, processing different graphs requires a strong reconfigurability of the device, achievable by altering the links (amplitudes and phases) between crystals and rerouting single photons in linear circuits. Implementing such a lattice in bulk optics, even for a smallscale demonstration, remains experimentally challenging.
We demonstrate an integrated graphbased quantum device with VLSI silicon photonics (Fig. 1b,d). We call it ‘Boya’. It is a synthetic twodimensional 4 × 4 lattice that consists of an array of spontaneous fourwave mixing integrated photonpair sources and a network of programmable linear optical waveguide circuits. The device directly enables complexweighted undirected graphs with eight vertices. Each pathway of single photons from one source to one detector represents a vertex. Each photonpair source connects two separate pathways and represents an edge. Quantum correlations in the device (Fig. 1b), emerging at sources and sharing between pathways, corresponds to the pairwise relation in the graph (Fig. 1c). An example is identified by two coloured pathways (vertices) and a source (edge). Moreover, connections between vertices can be altered by reconfiguring the waveguide circuits; amplitudes and phases of edges can be individually controlled by an array of key switches before sources and phase shifters before quantum erasers, respectively (Supplementary Table 1).
The VLSI graph quantum device was fabricated inhouse on a 200 mm silicononinsulator wafer by 180 nm complementary metal–oxide–semiconductor processes. Figure 1d illustrates a photograph of the full wafer. Each wafer contains 30 dies, each die contains four devices with slightly different designs, and each device integrates 2,446 components in a 12 mm × 15 mm footprint (Fig. 1d, white box). Waferscale charaterizations of the propagation loss are shown in Supplementary Fig. 4. A device has an array of 32 spontaneous fourwave mixing sources, forming a 4 × 4 grid, and each pair of adjacent sources produces one pair of degenerate single photons by reverse Hong–Ou–Mandel (RHOM) interference within a Mach–Zehnder interferometer^{18} (Fig. 1b, green highlights). The RHOM fringes for all the sources exhibit high contrast (Fig. 1g). The device contains 216 reconfigurable phase shifters (Supplementary Fig. 6d shows the characterizations). Four eightmode reconfigurable linear optical circuits (mean fidelity, 0.925(32); Fig. 1f and Supplementary Fig. 7) and a network of 463 ultralowloss waveguide crossers (mean loss, 0.038(4) dB; Fig. 1e) are used to reroute photons across the various paths. The device is fully optically and electrically packaged, and accessed by 100 optical inputs/optical outputs and 432 electronic inputs. With VLSI silicon photonics, the device here is the largestscale integrated quantum photonic device to date, to the best of our knowledge^{19,20,21}.
In the experiment, amplitudes and phases of the complexweighted edges of graphs (that is, the complex elements of graph’s adjacency matrices) are fully controlled. The connectivity of edges was altered by reconfiguring the eightmode Mach–Zehnder interferometer meshes (Fig. 1b), whose characterizations (imaginary part is not included) are reported in Fig. 1f. The real and imaginary parts of the two graphs’ edges are shown in Fig. 3b,i. Photon pairs were produced at the sources with a probability of 3.0%–4.5%, dependent on the pump power. We tested the quantum interference of indistinguishable photons by performing a heralded RHOM measurement between separate nonlinear sources (Fig. 1h). A postfiltering process was used to improve the spectral purity of photon sources at the cost of photon counts. The characterization and analysis of purity and indistinguishability are provided in Supplementary Section 4. Photons were detected by multichannel fibrecoupled superconducting nanowire singlephoton detectors. Multiphoton correlations, corresponding to the distributions of perfect matchings of graphs, were recorded by a multichannel counting module. We observed a fourphoton rate of 20 mHz in a typical setting of the experiment.
The realization of the graph quantum device in integrated optics (Fig. 1b) offers unique advantages, compared with the bulk optical one (Fig. 1c). It can perfectly match the optical length of paths for all the photons routing along lithographically defined circuits, ensuring good temporal mode matching. All the sources and circuits are monolithically integrated, enabling the reliable processing of graphs. Instead of pinning sources to the 4 × 4 grid, we here flatten this twodimensional grid into a onedimensional array, transposing the device structure into a braiding of waveguide circuits. This, thus, forms a synthetic graph lattice. It overcomes the problem of loss accumulation on pump and single photons, forced to pass through a series of sources^{22} (even for microring sources^{23}), which causes circuitdepthdependent edge amplitudes. It also avoids the complex demultiplexing and remultiplexing of photons with different colours. Moreover, to ensure the global coherence of the device, the whichsource information of all the single photons along every pathway must be coherently erased (Fig. 1b). The erasure process based on postselection ensures quantum coherence at the cost of photon counts, although by collecting all the outputs, the number of graphs’ vertices can be greatly enlarged (Supplementary Figs. 15 and 16 show the theoretical and experimental results).
We consider a general case of m × m quantum device, described by graph G with 2m vertices and m^{2} edges. The n pairs of single photons are created at n sources (a maximum of one pair per source, that is, weakly squeezed light source) and detected by 2m singlephoton detectors at 2m output modes (n ≤ m). The distributions of 2nfold coincidences are measured in the basis of {S_{1}, S_{2},…, S_{2m}}, where S_{i} denotes the number of photons in the ith mode. We consider no more than one photon click at each detector, that is, S_{i} = {0, 1}. Each 2nfold output of the 2n × 2n sublattice represents a perfect matching of subgraph G_{s} with 2n vertices, where the subscript ‘s’ denotes the subgraph. All the permutations of the 2nphoton outputs exactly represent a superposition of all the perfect matchings of G_{s}. Recording the multiphoton coincidences, thus, returns the probability distributions of the modulussquared hafnian or permanent of the adjacency submatrix O_{s}, that is, Prob(s) ∝ ∣Perm(O_{s})∣^{2} for a bipartite graph and Prob(s) ∝ ∣Haf(O_{s})∣^{2} for a general graph. In this context, similar to the standard quantum boson sampling^{24,25,26,27,28}, our device can return the sampling results from its output distribution, which, in general, has an exponentially large number of outcomes. An estimation of a single probability typically requires an exponentially increased running time to obtain a small error^{28}.
Creating genuine multiphoton multidimensional entanglement
Integrated quantum optics chips have allowed the generation of 15dimensional Bell state^{29}, fourqutrit hyperentangled cluster state^{30}, fourqubit graph state^{23,31} and eightqubit hyperentangled graph state^{32}. Entangling multiple particles in multiple dimensions could provide the key resources for strong quantum correlation tests^{33} and quditbased quantum computing^{34}. Recently, the multidimensional multiphoton Greenberger–Horne–Zeilinger (GHZ) states with three photons across three dimensions have been demonstrated in bulk optical setups, which are designed by machine learning techniques^{35}. However, the onchip generation of multidimensional multiphoton entangled states remains experimentally exclusive.
We target a general multidimensional multiphoton state \({\left\vert {{{\rm{GHZ}}}}\right\rangle }_{d}^{n}\) = \(\frac{1}{\sqrt{d}}\mathop{\sum }\nolimits_{k = 0}^{d1}{\left\vert k\right\rangle }^{\otimes n}\), where n is the number of photons, d is the local dimension of each photon and k〉 is the logical state in the kth mode. Reconfiguring a sublattice (part of Fig. 1b), we create different \({\left\vert {{{\rm{GHZ}}}}\right\rangle }_{d}^{n}\) entangled states. As an example, we discuss how to generate the \({\left\vert {{{\rm{GHZ}}}}\right\rangle }_{3}^{4}\) state in a 2 × 3 sublattice (Fig. 2a), whose configuration is given by the graph. The success of detecting fourfold coincidences only occurs in three indistinguishable processes in the device. Physically, two pairs of photons emerge either at sources {00, 00}, {11, 11} or {22, 22} (indicated on the sources), each representing the logical basis. Mathematically, these correspond to the three perfect matchings of the graph shown in Fig. 2a. Hence, as a result of the quantum interference of the three indistinguishable processes, we obtain the coherent entangled state \({\left\vert {{{\rm{GHZ}}}}\right\rangle }_{3}^{4}=(\left\vert 0000\right\rangle +\left\vert 1111\right\rangle +\left\vert 2222\right\rangle )/\sqrt{3}\). To verify the entanglement structures, we adopt an entanglement witness based on the Schmidt rank vector^{36}.
Figure 2a shows an estimated density matrix of the \({\left\vert {{{\rm{GHZ}}}}\right\rangle }_{3}^{4}\) state. The density matrix ρ is partially reconstructed by measuring 81 diagonal elements in the computational basis \({\hat{\varLambda }}_{i}^{\otimes n}\) and three offdiagonal terms in a coherent basis \({\hat{\varOmega }}_{\theta }^{\otimes n}={(\cos \theta {\hat{\varLambda }}_{i}+\sin \theta {\hat{\varLambda }}_{j})}^{\otimes n}\), where \({\hat{\varLambda }}_{i = 1}^{8}\) is the Gell–Mann basis. Note that quantum erasers (Fig. 1b) also work as arbitrary local projectors for qutrit states and thus allow fullstate measurement. We used an array of 12 detectors to record fourfold coincidence counts. From computational basis measurements (diagonal of ρ), we could unambiguously confirm the existence of three perfect matchings, in good agreement with Haf(O) = 3. We define a classical statistical overlap \(\gamma ={\sum }_{i}\sqrt{{p}_{i}{q}_{i}}\) to characterize the results, where p_{i} and q_{i} are the experimental and theoretical distributions, respectively; we obtain a γ value of 0.937. Coherence measurement results of \(\langle {\hat{\varOmega }}_{\theta }^{\otimes n}\rangle\) are provided in Supplementary Fig. 10. We estimate the quantumstate fidelity F = 〈ψ_{0}∣ρ∣ψ_{0}〉 from the density matrix, where ψ_{0}〉 is the ideal pure state. For the general state \({\left\vert {{{\rm{GHZ}}}}\right\rangle }_{d}^{n}\), if the value of F is larger than (d − 1)/d, it is genuinely entangled in the dimension of d (Supplementary Section 5)^{36}. We measure the fidelity of \({\left\vert {{{\rm{GHZ}}}}\right\rangle }_{3}^{4}\) to be 0.72(2), greater than the lower bound of 2/3. We have, therefore, verified genuine fourphoton GHZ entanglement in at least three dimensions, that is, having the Schmidt rank vector of {3, 3, 3, 3}.
We also present experimental results for other genuinely entangled states: GHZ{2, 2, 2, 2} in a 2 × 2 sublattice (Fig. 2b), GHZ{3, 3, 3} in a 2 × 3 sublattice (Supplementary Fig. 11) and Bell{3, 3} in a 1 × 3 sublattice (Fig. 2c; it represents the simplest twovertex graph^{29,37}). Figure 2d reports the experimental results for another class of multipartite entangled state of \({\left\vert {{{{W}}}}\right\rangle }_{3}=(\left\vert 100\right\rangle +\left\vert 010\right\rangle +\left\vert 001\right\rangle )/\sqrt{3}\). The measured distribution (diagonal of ρ) implies a number of three perfect matchings, consistent with Haf(O) = 3, and it shows a γ value of 0.957. If the fidelity is more than 2/3, it verifies the presence of entanglement. We obtain the fidelity of 0.727(19) and thus confirm the generation of a genuine threephoton W state. Moreover, tracing out one photon of W〉_{3}, the remaining photons are still partially entangled. The photonloss robustness of the W state is illustrated in Fig. 2e.
Measuring the perfect matchings of complex graphs
We then reprogram the entire graphbased quantum device to measure the probability distributions of the modulussquared permanent and hafnian matrix functions for more complex graphs. This task translates to the estimations of the number of perfect matchings of the corresponding bipartite graph and a general graph, respectively, that can be directly read out from the multiphoton coincidence patterns at the output of the quantum device.
Figure 3 shows the topologies of bipartite and general graphs, device configurations and corresponding measured perfect matching distributions. Their adjacency matrices are given in Supplementary Section 6. Figure 3b,i depicts the experimentally prepared bipartite and general graphs, in which the edge amplitudes and phases are indicated by thickness and colour, respectively. Edge’s amplitudes of the two graphs are randomly set, and the edge’s phases of the bipartite graph are set as {0, π} to indicate the quantum interference of processes, whereas phases of the general graph are randomly chosen to showcase the capability of realizing arbitrary graphs. There are a total of 70 permutations of fourphoton coincidences for all the submatrices, corresponding to all the submatrices and subgraphs. We collected N = 5,000 and 7,000 events in the bipartite and general graph experiments, with 68 and 120 h of data collection, respectively. Figure 3c,j reports the measured probability distributions for the bipartite and general graphs, respectively, which are in good agreement with the theoretical predictions by classically calculating ∣Per(O_{s})∣^{2} and ∣Haf(O_{s})∣^{2} for every submatrix. To characterize the results, we use the statistical overlap \(\gamma =\sum \sqrt{{p}_{i}{q}_{i}}\) and Kolmogorov distance D = ∑∣p_{i} − q_{i}∣/2, where p_{i} and q_{i} are the theoretical and experimental probabilities, respectively. Ideally, we have γ = 1 and D = 0. In the experiment, we obtain γ = 0.979(2) and D = 0.122(7) for permanent (Fig. 3c) and γ = 0.986(2) and D = 0.098(7) for hafnian (Fig. 3j). Nonperfect indistinguishability of the sources and processes has been taken into consideration in theory^{38,39,40} (Supplementary Sections 4.4 and 6.1). A set of measurements for a doublelayered graph is reported in Supplementary Fig. 16.
The implementation of quantum graph processing requires an efficient approach for the validation of experimental outcomes. Validation helps to rule out other possible hypotheses. We compare the experimental results with those having an input of distinguishable photons, as well as a uniform weighted graph (uniform matrix). We adopt a Bayesian updating method, like that used for the validation of boson sampling experiments^{41}. In Fig. 3d,k, a Bayesian analysis is used to validate the experimental results for the bipartite and general graphs, against the hypothesis that photons generated from different nonlinear processes are distinguishable. The confidence of discrimination is higher than 99.99%, demonstrating the genuine quantum interference of multiphoton processes in the device. In Fig. 3e,i, we discriminate between our experimental data and those predicted by a uniform graph, and this hypothesis is also ruled out. Other validations using a likelihood ratio test are provided in Supplementary Fig. 12.
Moreover, another new protocol using the statistical analysis^{42,43} of the sampled datasets can efficiently rule out the hypothesis of distinguishable photons, and such a protocol has been adopted to validate boson sampling ^{44,45,46,47}. It is based on the twomode correlation function, defined as \({C}_{ij}=\langle {\hat{n}}_{i}{\hat{n}}_{j}\rangle \langle {\hat{n}}_{i}\rangle \langle {\hat{n}}_{j}\rangle\), where \(\hat{n}\) is the bosonic number operator and i, j indicate the two output modes of the devices. Figure 3f,m reports the experimental C dataset for the general and bipartite graphs that are in good agreement with the theoretical results. We place the experimental data point into a (CV, S) plane (CV is the coefficient of variation and S is the skewness), to discriminate it with the cloud of indistinguishable photons rather than the cloud of distinguishable photons. The clouds are numerically obtained from 5,000 random graphs. The results are shown in Fig. 3g,n. The results of the histogram of C dataset and the statistical analysis in a (CV, NM) plane (NM is the normalized mean) are provided in Supplementary Fig. 13. This statistical analysis, thus, confirms the genuine manyphoton quantum interference in the graphbased quantum device.
We have reported a reconfigurable graphbased quantum device on the VLSI photonic chip, and we reprogrammed it to execute diverse tasks defined by graphs. The device was reconfigured to show the generation, manipulation and certification of genuine multiphoton multidimensional entanglement, which could provide a key resource for universal multidimensional quantum computing^{34} and information processing in the future^{48}. Using this approach, it could create multidimensional multiphoton GHZ states with an arbitrary dimension, their heralded states and gates by algorithmic optimization^{10,11}. We reconfigured the device to map abstract general graphs onto the physical hardware, and estimated these graphs’ perfect matchings by quantum measurements.
Interestingly, akin to quantum boson samplers^{24,25,26,27,28} that have recently shown quantum computational advantages^{45,46,47}, as well as photonic quantum annealers^{49,50} that can simulate large spin models, the graph quantum devices also seek the results of graph or matrixassociated problems (Supplementary Section 7). The graphbased quantum device benchmarked in this work represents a general linear optical quantum device that can be arbitrarily reprogrammed to implement many diverse tasks in quantum information processing. Adopting a graph’s highlevel visualizability and powerful mathematical machinery, it could provide a versatile hardware platform, such as to engineer complex quantum entanglement^{8,9}, design new quantum gates and resources states^{11}, learn complex quantum systems^{51,52} and train quantum processors^{10,53}. With advanced silicon photonic quantum devices and technologies, for example, highly pure and indistinguishable integrated photonpair sources^{54}, lowloss sources with wavelength (de)multiplexers that can be embedded in a twodimensional mesh, largescale integration of highefficiency singlephoton detectors^{55} and waferscale integration of photonic circuits^{56} (Supplementary Section 8 provides the analysis and calculation), there is tremendous latent promise of complex graphtheoretical quantum processing for potential interesting quantum applications^{57,58}.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.
References
Raussendorf, R. & Briegel, H. J. A oneway quantum computer. Phys. Rev. Lett. 86, 5188–5191 (2001).
Walther, P. et al. Experimental oneway quantum computing. Nature 434, 169–176 (2005).
Harris, N. C. et al. Quantum transport simulations in a programmable nanophotonic processor. Nat. Photon. 11, 447–452 (2017).
Ehrhardt, M. et al. Exploring complex graphs using threedimensional quantum walks of correlated photons. Sci. Adv. 7, eabc5266 (2021).
Qiang, X. et al. Implementing graphtheoretic quantum algorithms on a silicon photonic quantum walk processor. Sci. Adv. 7, eabb8375 (2021).
Cabello, A., Severini, S. & Winter, A. Graphtheoretic approach to quantum correlations. Phys. Rev. Lett. 112, 040401 (2014).
Perseguers, S. et al. Quantum random networks. Nat. Phys. 6, 539–543 (2010).
Krenn, M., Gu, X. & Zeilinger, A. Quantum experiments and graphs: multiparty states as coherent superpositions of perfect matchings. Phys. Rev. Lett. 119, 240403 (2017).
Gu, X. et al. Quantum experiments and graphs II: quantum interference, computation, and state generation. Proc. Natl Acad. Sci. USA 116, 4147–4155 (2019).
Krenn, M. et al. Conceptual understanding through efficient automated design of quantum optical experiments. Phys. Rev. X 11, 031044 (2021).
RuizGonzalez, C. et al. Digital discovery of 100 diverse quantum experiments with PyTheus. Preprint at arXiv https://doi.org/10.48550/arXiv.2210.09980 (2022).
Valiant, L. The complexity of computing the permanent. Theor. Comput. Sci. 8, 189–201 (1979).
Pan, J.W. et al. Multiphoton entanglement and interferometry. Rev. Mod. Phys. 84, 777–838 (2012).
Slussarenko, S. & Pryde, G. J. Photonic quantum information processing: a concise review. Appl. Phys. Rev. 6, 041303 (2019).
Flamini, F., Spagnolo, N. & Sciarrino, F. Photonic quantum information processing: a review. Rep. Prog. Phys. 82, 016001 (2018).
Mandel, L. Coherence and indistinguishability. Opt. Lett. 16, 1882–1883 (1991).
Krenn, M. et al. Entanglement by path identity. Phys. Rev. Lett. 118, 080401 (2017).
Silverstone, J. W. et al. Onchip quantum interference between silicon photonpair sources. Nat. Photon. 8, 104–108 (2013).
Wang, J. et al. Integrated photonic quantum technologies. Nat. Photon. 14, 273–284 (2020).
Elshaari, A. W. et al. Hybrid integrated quantum photonic circuits. Nat. Photon. 14, 285–298 (2020).
Pelucchi, E. et al. The potential and global outlook of integrated photonics for quantum technologies. Nat. Rev. Phys. 4, 194–208 (2022).
Feng, L.T. et al. Onchip quantum interference between the origins of a multiphoton state. Optica 10, 105–109 (2023).
Llewellyn, D. et al. Chiptochip quantum teleportation and multiphoton entanglement in silicon. Nat. Phys. 16, 148–153 (2020).
Aaronson, S. & Arkhipov, A. The computational complexity of linear optics. In Proc. FortyThird Annual ACM Symposium on Theory of Computing 9, 143–252 (ACM, 2013).
Bentivegna, M. et al. Experimental scattershot boson sampling. Sci. Adv. 1, e1400255 (2015).
Hamilton, C. S. et al. Gaussian boson sampling. Phys. Rev. Lett. 119, 170501 (2017).
Kruse, R. et al. Detailed study of Gaussian boson sampling. Phys. Rev. A 100, 032326 (2019).
Brádler, K. et al. Gaussian boson sampling for perfect matchings of arbitrary graphs. Phys. Rev. A 98, 032310 (2018).
Wang, J. et al. Multidimensional quantum entanglement with largescale integrated optics. Science 360, 285–291 (2018).
Reimer, C. et al. Highdimensional oneway quantum processing implemented on dlevel cluster states. Nat. Phys. 15, 148–153 (2019).
Adcock, J. C. et al. Programmable fourphoton graph states on a silicon chip. Nat. Commun. 10, 3528 (2019).
Vigliar, C. et al. Errorprotected qubits in a silicon photonic chip. Nat. Phys. 17, 1137–1143 (2021).
Friis, N. et al. Entanglement certification from theory to experiment. Nat. Rev. Phys. 1, 72–87 (2019).
Chi, Y. et al. A programmable quditbased quantum processor. Nat. Commun. 13, 1166 (2022).
Erhard, M. et al. Experimental Greenberger–Horne–Zeilinger entanglement beyond qubits. Nat. Photon. 12, 759–764 (2018).
Huber, M. & de Vicente, J. I. Structure of multidimensional entanglement in multipartite systems. Phys. Rev. Lett. 110, 030501 (2013).
Kysela, J. et al. Path identity as a source of highdimensional entanglement. Proc. Natl. Acad. Sci. USA 117, 26118–26122 (2020).
Tichy, M. C. Sampling of partially distinguishable bosons and the relation to the multidimensional permanent. Phys. Rev. A 91, 022316 (2015).
Renema, J. J. et al. Efficient classical algorithm for boson sampling with partially distinguishable photons. Phys. Rev. Lett. 120, 220502 (2018).
Spagnolo, N. et al. Experimental validation of photonic boson sampling. Nat. Photon. 8, 615–620 (2014).
Paesani, S. et al. Generation and sampling of quantum states of light in a silicon chip. Nat. Phys. 15, 925–929 (2019).
Walschaers, M. et al. Statistical benchmark for boson sampling. New J. Phys. 18, 032001 (2016).
Phillips, D. S. et al. Benchmarking of Gaussian boson sampling using twopoint correlators. Phys. Rev. A 99, 023836 (2019).
Giordani, T. et al. Experimental statistical signature of manybody quantum interference. Nat. Photon. 12, 173–178 (2018).
Zhong, H.S. et al. Quantum computational advantage using photons. Science 370, 1460–1463 (2020).
Zhong, H.S. et al. Phaseprogrammable Gaussian boson sampling using stimulated squeezed light. Phys. Rev. Lett. 127, 180502 (2021).
Madsen, L. S. et al. Quantum computational advantage with a programmable photonic processor. Nature 606, 75–81 (2022).
Erhard, M., Krenn, M. & Zeilinger, A. Advances in highdimensional quantum entanglement. Nat. Rev. Phys. 2, 365–381 (2020).
Marandi, A. et al. Network of timemultiplexed optical parametric oscillators as a coherent Ising machine. Nat. Photon. 8, 937–942 (2014).
Inagaki, T. et al. Largescale Ising spin network based on degenerate optical parametric oscillators. Nat. Photon. 10, 415–419 (2016).
FlamShepherd, D. et al. Learning interpretable representations of entanglement in quantum optics experiments using deep generative models. Nat. Mach. Intell. 4, 544–554 (2022).
Wang, J. et al. Experimental quantum Hamiltonian learning. Nat. Phys. 13, 551–555 (2017).
Banchi, L., Quesada, N. & Arrazola, J. M. Training Gaussian boson sampling distributions. Phys. Rev. A 102, 012417 (2020).
Paesani, S. et al. Nearideal spontaneous photon sources in silicon quantum photonics. Nat. Commun. 11, 2505 (2020).
Pernice, W. H. P. et al. Highspeed and highefficiency travelling wave singlephoton detectors embedded in nanophotonic circuits. Nat. Commun. 3, 1325 (2012).
Zhang, X. et al. A largescale microelectromechanicalsystemsbased silicon photonics lidar. Nature 603, 253–258 (2022).
Huh, J. et al. Boson sampling for molecular vibronic spectra. Nat. Photon. 9, 615–620 (2015).
Banchi, L. et al. Molecular docking with Gaussian boson sampling. Sci. Adv. 6, eaax1950 (2020).
Acknowledgements
We thank L. Jin, H. Zhao, and J. Feng from the United Microelectronics Center in Chongqing (CUMEC) for experimental assistance. We acknowledge I. Faruque from the University of Bristol, M. Krenn (currently in Max Planck Institute) and M. Erhard from Vienna University, and C. Y. Lu and X. Gu from the University of Science and Technology of China for useful discussions. We acknowledge support from the Natural Science Foundation of China (nos. 62235001, 61975001, 62274179, 62125503, 91950205, 61961146003, 12125402, 11975026), the Innovation Program for Quantum Science and Technology (no. 2021ZD0301500), the National Key R&D Program of China (nos. 2019YFA0308702, 2018YFA0704404, 2022YFB2802400), Beijing Natural Science Foundation (Z190005, Z220008) and Key R&D Program of Guangdong Province (2018B030329001). D.D. acknowledges support from the National Science Fund for Distinguished Young Scholars (61725503), the Fundamental Research Funds for the Central Universities and the Leading Innovative and Entrepreneur Team Introduction Program of Zhejiang (2021R01001). S.P. acknowledges funding from the Cisco University Research Program Fund no. 2021234494. J.W.S. acknowledges the generous support from the Leverhulme Trust (ECF2018276) and the UKRI (MR/T041773/1). A.L. acknowledges support from the EPSRC Hub in Quantum Computing and Simulation (EP/T001062/1). Y.D. acknowledges support from the Villum Fonden Young Investigator project QUANPIC (ref. 00025298) and Danish National Research Foundation Center of Excellence, SPOC (ref. DNRF123).
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J.W. conceived the idea and initiated the project. J.B., Z.F., T.P., J.M. and Y. Chi equally contributed to this work (J.B. and Z.F. characterized and measured the devices; J.B., Z.F. and T.P. analysed the entanglement and matrix functions data; and J.M. and Y. Chi fabricated the devices). J.B., Z.F., J.M., Y.M., T.D., L.Z. and Y.Z. implemented the experiment. C.V., S.P., H.H., R.S., J.W.S. and J.W. implemented some early measurements. J.M., Y. Chi, B.T., Y.Y., A.L.M. and Y.D. fabricated the device. J.B., Z.F., Y. Cao, C.Z., T.D., X.C., Y.P., D.L., D.D. and J.W. designed the devices. J.B., Z.F., T.P., Y. Cao, C.Z., X.J. and Q.H. performed the theoretical analysis. D.D., Z.L., J.L., W.W., A.L., L.K.O., M.G.T., J.L.O., Y.D., Q.G. and J.W. managed the project. J.B., Z.F., T.P., J.M., Y. Chi and J.W. wrote the manuscript, with input from all the authors.
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Supplementary Figs. 1–16, Sections 1–8 and Tables 1 and 2.
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Bao, J., Fu, Z., Pramanik, T. et al. Verylargescale integrated quantum graph photonics. Nat. Photon. 17, 573–581 (2023). https://doi.org/10.1038/s4156602301187z
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DOI: https://doi.org/10.1038/s4156602301187z
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