Abstract
Future quantum computers require a scalable architecture on a scalable technology—one that supports millions of highperformance components. Measurementbased protocols, using graph states, represent the state of the art in architectures for optical quantum computing. Silicon photonics technology offers enormous scale and proven quantum optical functionality. Here we produce and encode photonic graph states on a massmanufactured chip, using four onchipgenerated photons. We programmably generate all types of fourphoton graph state, implementing a basic measurementbased protocol, and measure highvisibility heralded interference of the chip’s four photons. We develop a model of the device and bound the dominant sources of error using Bayesian inference. The combination of measurementbased quantum computation, silicon photonics technology, and onchip multipair sources will be a useful one for future scalable quantum information processing with photons.
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Introduction
Graph states are key entangled resources for quantum information processing. They are quantum states, which can be drawn as a graph, with a qubit on each vertex and pairwise entanglement on each edge^{1}. In measurementbased quantum computing, where singlequbit measurements on a graph state drive the computation forward, particular graphs enable particular computational tasks^{2}. Topological quantum error correction, relying centrally on graph states, will provide essential noise tolerance to future experimental realisations^{3}. Graph states also play a central role as platforms for the simulation of complex processes and dynamics^{4}, and for quantum secret sharing protocols^{5}. As such, graph states have featured strongly in experiment, in both optics^{6,7,8,9} and other platforms^{10}. The reconfigurable generation of arbitrary graphs, never before achieved in optics, will accelerate development of many graphbased applications.
Integrated optics promises new levels of scale for optical quantum devices. It offers robustly modematched, miniature components, lithographically defined in a planar process. Phase stability and matched optical path lengths are guaranteed. Stateoftheart chipscale devices now exhibit loss and error performance approaching that of bulk and fibre systems. Quantum optical functionality has been demonstrated in all major technology platforms: lithium niobate^{11}, silica^{12,13,14} (both lithographic and laserwritten), silicon nitride^{15}, gallium arsenide^{16}, indium phosphide^{17}, and silicon^{18,19}.
Silicon devices have rapidly grown in complexity in recent years, with quantum demonstrators now exceeding 500 onchip components^{20}, and classical silicon photonic devices having thousands^{21,22}. Integration with CMOS electronics could push this scale further still, by miniaturising control and interconnect functionality^{22}. A quantum device’s computational power is related to the quantum configuration (Hilbert) space accessible to it. In optics, this space has m^{n} dimensions, for n photons scattered across m modes. So far, the scaling up of silicon quantum photonics has mainly involved scattering one or two photons (n = 1 or 2) over more and more waveguides (increasing m) as a route to polynomially larger Hilbert spaces^{20,23}. Extending chipscale quantum optics into the multipair regime, increasing n, is a crucial step for exponential Hilbert space scaling. Only recently has onchip heralded interference between onchipgenerated photons been demonstrated^{24,28}, though visibility is limited and no quantum information has yet been encoded.
We present a silicon quantumoptical device that can generate four photons and programmably encode them into either of the two classes^{25} of fourqubit graph state entanglement—classes closed under local unitary transformations. We refer to these two classes by their bestknown members: ‘star’ S_{4}〉, and ‘line’ L_{4}〉. We observe quantum interference between photons heralded from two of the onchip photonpair sources, characterise the stabilisers of the star and line states, and test their multipartite nonlocality. Finally, we use Bayesian inference to access key device parameters, based on the fourphoton data alone.
Results
Device and experiment design
Our device, shown schematically in Fig. 1, operates in four stages. (1) Four photons in two pairs are generated in superposition over four sources. (2) These are demultiplexed by wavelength and rearranged to group signal and idler photons. The resulting dualrail, pathencoded qubit state is a product of Bellpairs, Φ^{+}〉_{1,3} ⊗ Φ^{+}〉_{2,4} (with qubit indices in subscript). (3) The signalphoton qubits are operated upon by a reconfigurable postselected entangling gate (RPEG). This can be programmed to perform either a fusion or controlledZ operation, to generate star or linetype entanglement^{25}, respectively, with postselected probability 1/2 or 1/9. (4) We then perform arbitrary singlequbit projective measurements, using Mach–Zehnder interferometers (MZI), on the fourqubit states. A full description of the state evolution can be found in Supplementary Note 3 and Supplementary Fig. 5.
The χ^{(3)} process, spontaneous fourwave mixing, converts bright telecommunicationsband pump pulses into quantumcorrelated signal and idler photons in the spiralled silicon waveguides of our source stage^{26}. Thermooptic phase modulators provide electronic reconfigurability throughout the device. Focussing vertical grating couplers connect onchip waveguides to optical fibre. Finally, signal and idler photons are tightly filtered in fibre (pump:photon filtering bandwidth ratio 2:1, see Supplementary Fig. 7), and registered by superconducting nanowire singlephoton detectors. See Methods for more details. Using this apparatus, we measure heralded twophoton fringes, the purity of our sources, and the stabilisers of our programmed graph states with four photons.
Heralded HongOuMandel interference
Indistinguishable photons are key for highfidelity operation. HongOuMandel (HOM) interference, whereby two photons launched into the two ports of a beamsplitter bunch at the outputs, directly indicates their distinguishability—over all degrees of freedom—via the residual rate of antibunching from the beamsplitter outputs. When the interfering photons are heralded from entangled pairs (four photons total), nonunit photon purity also contributes to their distinguishability. Onchip path lengths are naturally matched, so rather than using the conventional timedelay HOM dip, we measure an onchip heralded fringe^{24,27} (Supplementary Note 2 and Supplementary Fig. 4 relate these two measurements). In both measurements, the residual antibunching rate indicates the photons’ overall distinguishability. We interfere signal photons from sources 2 and 3, heralding on the two corresponding idler photons. By tuning the central RPEG Mach–Zehnder’s phase, ϕ, we sweep its effective reflectivity, R(ϕ), from R(0) = 0, through R(π) = 1, to R(2π) = 0. At R(mπ), there is no interference, while at R(mπ + π/2) = 1/2, the HongOuMandel effect occurs \((m \in {\Bbb Z})\). The measured fringe, shown in Fig. 2d, exhibits a visibility of V = 0.82 ± 0.02, in line with other measurements on chip^{24,28}. Here, V = (N_{max }− N_{min})/(N_{max} + N_{min}), and N_{max} and N_{min} are the maximum and minimum values of the fitted sinusoid; classical light is limited to V < 1/3. The conventional HOMdipequivalent visibility, upperbounded by the heralded purity of our photons, is V_{HOM} = (N_{max }− 2N_{min})/N_{max }= 0.80 ± 0.02 (see Supplementary Note 2). The photonpair generation probability here is p = 0.06. We corroborate V_{HOM} by measuring the unheralded secondorder correlation function g^{(2)}(0) for the eight modes of our four onchip sources, implying^{29} heralded purities between 0.82 and 0.92 (Supplementary Note 1 and Supplementary Fig. 2 contain a full listing). New onchip parametric source designs will improve brightness and purity further^{14,30,31}.
Graph state measurements
We verify the generation of the fourphoton star and line graph states (S_{4}〉 and L_{4}〉) by measuring their 16 stabilisers^{32}, g_{{i}}, where {i} is the set of generators whose product composes each stabiliser (e.g., g_{12} = g_{1}g_{2}). The four stabiliser generators of the state S_{4}〉 are:
where X, Y, and Z are Pauli matrices and I is the identity matrix; tensor products are implied. For L_{4}〉, the stabiliser generators are:
Measurements for each stabiliser are plotted in Fig. 2a, b, for S_{4}〉 and L_{4}〉, respectively. From these, we compute fidelities, shown in Table 1, and find that both states robustly satisfy the F > 1/2 threshold to witness genuine multipartite entanglement^{32}. These fidelities compare favourably with the first bulkoptics measurements on these states^{6,33}. In these and subsequent fourphoton measurements, we reduce the photonpair generation probability to p = 0.03 to suppress multiphoton noise.
We perform a basic measurementbased protocol^{34} by projecting various qubits of S_{4}〉 onto 0〉, and measuring the remaining two and threequbit graph states. We denote these states \(\left {S_4} \right\rangle _J \, = ( \otimes _{j \notin J}\langle 0_j)S_4\rangle\), where J is the set of remaining (unprojected) qubits. The threequbit state S_{4}〉_{1,3,4} and the twoqubit states S_{4}〉_{1,4} and S_{4}〉_{3,4} can be produced by projecting qubits {2}, {2, 3}, and {1, 2} onto 0〉, respectively. Measured fidelity data for these states, along with those for the two input Bellpairs, Φ^{+}〉_{1,3} and Φ^{+}〉_{2,4}, are listed in Table 1. Notice that the two photons encoding S_{4}〉_{1,4} are orthogonal in colour, and have never interacted.
Tests of multipartite nonlocality
Mermin tests let us verify the nonlocality of multipartite states^{35,36}. We construct tests^{32} comprising two and three measurement settings per qubit, \({\mathscr{M}}_{II}^G\) and \({\mathscr{M}}_{III}^G\), based on the stabiliser observables of each graph state G. For convenience, we use \({\mathscr{M}}_{II}^G\) and \({\mathscr{M}}_{III}^G\) to indicate both the test’s operator and the modulus of its expectation value (e.g., \(\langle {\mathscr{M}}_{II}^G\rangle \)). Results are listed in Table 1 and plotted in Fig. 2c. \({\mathscr{M}}_{II}^G\) allows a choice, one for each graph symmetry, of stabilisers; we report only the optimal choice here, though all choices exceed the classical bound. Other measurement results are reported in Supplementary Table 1. We find that S_{4}〉 exceeds both \({\mathscr{M}}_{II}^G \, < \, 2\) and \({\mathscr{M}}_{III}^G \, < \, 12\) classical bounds. L_{4}〉 exceeds the classical bound for \({\mathscr{M}}_{II}^G\), but not for the more strict \({\mathscr{M}}_{III}^G\). The higher postselection penalty of the controlledZ, required to generate L_{4}〉, results in a decreased fidelity, of which \({\mathscr{M}}_{III}^G\) is a simple rescaling.
Understanding device performance
As quantum devices increase in complexity, the scaling of errors is of critical importance. The first step to correcting any error is to understand its source. Error models differ substantially between platforms, even within optics. Here, we develop methods for quantifying lowlevel performance parameters and apply these methods to our device. We seek to understand the effects of photon distinguishability, multiphoton noise, and thermooptic phase error. Each effect is modelled independently. Since all effects contribute to the data, our estimates for each parameter are pessimistic. We apply Bayesian parameter estimation to learn the likeliest model parameters based on the fourphoton stabiliser data^{37}. The indistinguishability (σ), multiphoton emission (p), and random phase error (δ), are estimated with no prior assumptions. The resulting probability distributions of the three parameters are reported in Fig. 2e–g, for both S_{4}〉 and L_{4}〉. Fitting each with a normal distribution, we compute parameter estimates and standard deviations: σ_{S,L }= {0.82 ± 0.01, 0.82 ± 0.01}, p_{S,L }= {0.036 ± 0.009, 0.037 ± 0.012}, and δ_{S,L }= {0.185 ± 0.007 rad, 0.182 ± 0.009 rad}. Our other measurements (HOM interference, g^{(2)}, source brightness, and crosstalk—see Methods) are compatible with these estimates; the distributions for the two states, S_{4}〉 and L_{4}〉, also broadly agree. This approach can reveal additional device performance information from existing data—no new measurements are required.
To completely describe device performance a holistic error model—one that simultaneously captures all the effects—is needed. To formulate such a model requires knowledge of difficulttoaccess quantities and significant computational power. A distinguishability model, for example, must have the Schmidt spectrum of each source—inaccessible from simple HOM dips—and a common basis for them. Computationally, modelling variable, high photonnumber states in highdimensional spaces is a challenge. Moreover, the three effects we studied affected the observables in a similar way and depended on the state: a holistic model may not help to effectively distinguish these effects, but tailored or adaptive measurements may help.
Discussion
Highfidelity is crucial for many quantum information processing applications. We demonstrate entangled resources of sufficient fidelity to violate several Mermin inequalities, though future scaling will need higher fidelity still. Improved throughput (lower loss) increases fidelity: directly, multiphoton noise scales with loss, due to an increase in the relative likelihood of a multiphoton term being detected; and indirectly, shorter integration times, via higher throughput and increased rates, yield improved stability and so reduced variation in system parameters (phase setting error, calibration, pump, fibrechip coupling, detector efficiency, etc.). Using stateoftheart silicon photonics and customised fabrication processes, fourfold coincidence rates could be propelled to the 100kHz regime (see Supplementary Note 4). Finally, it should be noted that our observation of rates in the 1mHz range are comparable to rates observed in first experiments achieving historic increases in photon number (e.g., ref. ^{38}).
We have demonstrated a multiphoton, multiqubit capability using standard, commercially available silicon photonic components. The techniques we have demonstrated—combining multiple postselected Bellpairs in reconfigurable gates—can be applied to construct sophisticated chipscale graph state generators. Although four is the largest number of dualrail qubits for which all entanglement classes can be postselected, six and eightqubit devices can still access most classes: 10/11 and 73/101, respectively^{25}.
Although our postselectionreliant approach to sourcing photons and preparing entanglement is not scalable, scalable approaches (e.g., those using feedforward^{39,40,41}) must overcome many of the same challenges. We can now bring the reconfigurability and control of integrated photonics to bear on the exploration of multiphoton space. The combination of multiple photons and highdimensional techniques^{20} will soon make vast Hilbert spaces accessible. Ultimately, postselection lets us test the components and techniques key to unlocking the huge graph states needed for photonic quantum computation^{41,42}.
Graph states are, and will continue to be, a building block of largescale quantum technology. We have demonstrated a photonic generator of arbitrary graph states, in a miniature, highperformance technology. We have encoded quantum information in more than one pair of photons generated on a chip. Future increases in photon number depend principally on improving rates, by engineering photon throughput, and dispensing with postselection. This prototype represents the next step towards a future of largescale quantum photonic devices.
Methods
Experimental setup
Pump pulses at 1544.40 nm (1.1 ps pulse duration, 500 MHz repetition rate) from an erbiumdoped fibre laser (Pritel) are filtered with squareshaped, 1.4nmbandwidth filters and injected into the device. The average launched pump power is 4.5 mW. The pump pulse spectrum and autocorrelation are shown in Supplementary Fig. 3. The sech^{2} pulse duration is 4.80 ± 0.03 ps. Signal and idler photons are collected at pumpdetuned ±4.8 nm, and filtered with squareshaped, 0.7nmbandwidth filters (Opneti DWDM) for spectral shaping and pump light rejection. They are detected offchip by four superconducting nanowire singlephoton detectors with 80 ± 5% efficiency (Photon Spot), operating around 0.85 K. Timetags are generated (UQDLogic) and converted to coincidences by bespoke software. The device is mounted using thermal epoxy and wirebonded to an FR4 printed circuit board; temperature is stabilised using a closedloop thermoelectric cooler. Optical coupling to fibre is via a fibre Vgroove array (OZ Optics) and a 6axis piezoelectric actuator (Thorlabs). Analogue voltage drivers (Qontrol Systems) are used to drive the onchip phase shifters, with 16bit and 300μV resolution. The device was fabricated by the A*STAR Institute of Microelectronics, Singapore. A 220nm device layer performs waveguiding, atop a 2m buried oxide (silicononinsulator) with an oxide top cladding. It has an area of 1.4 × 3 mm^{2} with 500nmwide waveguides. Kilohertzbandwidth thermooptic phase modulators are formed by TiN heaters, 180 × 2 μm^{2}, positioned 2 μm above the waveguide layer. See Supplementary Fig. 1 for a schematic of the experimental setup.
Phase shifter calibration and crosstalk
We calibrate the device’s thermooptic phase shifters by illuminating their enclosing MZIs with a continuouswave laser at the relevant wavelength, and applying a range of voltages to produce a fringe at the MZI output. We fit this fringe with a function A sin(f ⋅ P(V) + ϕ_{0}) + c, where P(V) = I(V) ⋅ V is the Joule heating of the phase shifter, to find A, f, ϕ_{0}, and c. By measuring the currentvoltage relationship of the phase shifters and fitting them to I(V) = ρ_{1}V + ρ_{2}V^{2 }+ ρ_{3}V^{3}, we can ‘dial in’ a phase ϕ_{d} by numerically solving the quartic equation ϕ_{d} = f ⋅ I(V) ⋅ V + ϕ_{c}. Lossmatched, evanescently coupled waveguide taps with 2% transmission are strategically placed around the device to allow independent calibration of each onchip phase shifter.
We measure the phase deviation within one onchip demultiplexer per unit power dissipated in the other thermooptic modulators. A thermal crosstalk coefficient of 0.003 rad mW^{−1} results. The average power dissipated over all chip configurations used in the stabiliser measurements was 443 and 472 mW for the star and line states, respectively. These distributions indicate an average deviation from the mean of 39 and 22 mW for the two states. Working backwards, we estimate the average thermooptic phase error is 0.12 rad and 0.065 rad, respectively. Power histograms and crosstalk fringes are shown in Supplementary Fig. 6.
Loss
The device insertion loss is 26.1 dB for the light path through source 1 to the 0〉 output of qubit 1, after optimising the relevant phase settings. We estimate losses, based on measurements on test structures on the same die, as: 4 dB per vertical grating coupler, 0.65 dB per 2 × 2 multimode interferometer (MMI), 3 dB cm^{−1} of straight waveguide propagation, and 7.5 dB cm^{−1} of spiral waveguide propagation. All measurements are at 1544.4 nm. By including offchip losses (3 dB), input coupling (one grating, two MMIs), and one half of the source length, we estimate that signal photons experience a loss of 19.3 dB.
HOMfringe visibilities
In an ideal HOM fringe the maximum is twice the background ‘distinguishable’ level of an ideal HOM dip. To calculate the equivalent dip visibility V_{HOM} from the maximum and minimum values measured in a fringe, we use V_{HOM} = (N_{max}/2 − N_{min})/(N_{max}/2) = (N_{max }− 2N_{min})/N_{max}. More details are in Supplementary Note 2.
Measuring state fidelities
We wish to find the fidelity of our experimental state ρ_{ex}, with a graph state ρ, with stabilisers {g_{i}}. Since ρ is a stabiliser state, \(\rho = \frac{1}{{2^n}}\mathop {\sum}\nolimits_i^{2^n} {g_i}\). Hence, \(F = {\mathrm{tr}}[\rho _{{\mathrm{ex}}}\rho ] = \frac{1}{{2^n}}\mathop {\sum}\nolimits_i^{2^n} {{\mathrm{tr}}} [g_i\rho _{{\mathrm{ex}}}] = \frac{1}{{2^n}}\mathop {\sum}\nolimits_i^{2^n} {\langle g_i\rangle }\) (see ref. ^{32}). This measurement method is used for all reported state fidelities.
Local Pauli expectation values are measured by projecting each of the 2^{n} eigenvectors onto each qubit’s single output waveguide and counting nfold coincidences (in our experiment, n = 4). Summing the results of each projective measurement (total counts C_{j}) by eigenvalue and normalising gives \(\langle g_i\rangle = \mathop {\sum}\nolimits_j^{2^n} {\lambda _j} C_j/\mathop {\sum}\nolimits_j^{2^n} {C_j}\). Here the eigenvalue of stabiliser projector j is a product of its local components \(\lambda _j = \mathop {\prod}\nolimits_k^n \mu _j^{(k)}\), with \(\mu _j^{(k)} \in \{  1,1\}\) being the eigenvalue of the local operator on qubit k. Supplementary Note 5 contains a complete list of each state’s stabilisers.
Mermin tests
For both S_{4}〉 and L_{4}〉, we measure every twosetting Mermin test that can be composed from its stabilisers. The tests for the star state are as follows (graph symmetries are indicated by an arrow): \({\mathscr{M}}_{II}^S = g_4(1 + g_2g_3 + g_2g_1 + g_3g_1),g_4 \to g_4g_1\) and \({\mathscr{M}}_{II\prime }^S = g_4(1 + g_i)(1 + g_j),g_4 \to g_4g_k,\) where g_{i} are the stabiliser generators and i, j, k = {1, 2, 3}. For the line state: \({\mathscr{M}}_{II}^L = g_1(1 + g_2)(1 + g_3),\) with g_{2} → g_{2}g_{4} and \({\mathscr{M}}_{II{\prime}}^L = g_1(1 + g_3)(g_2 + g_4),\) with g_{2} → g_{2}g_{4}, and g_{i} → g_{i}g_{i+1}, for i ∈ {1, 2, 3, 4}. Localrealistic (‘classical’) theories obey \({\mathscr{M}}_{II}^G \, < \, 2,\) while \({\mathscr{M}}_{II}^G \, < \, 4\) for quantum mechanics.
We also report a threesetting Mermin test: \({\mathscr{M}}_{III}^G = \mathop {\sum}\nolimits_i {\langle g_i\rangle } ,\) where the sum is take over all the 2^{n} (16) stabilisers of the graph state. Localrealistic theories obey \({\mathscr{M}}_{III}^G \, < \, 12,\) while \({\mathscr{M}}_{III}^G \, < \, 16\) for quantum mechanics.
Bayesian parameter estimation
We use three independent models to simulate the effects of partial distinguishability, multiphoton emission, and phase error (see Supplementary Note 6 for model details). These output a fourfold rate for each measurement setting, used to estimate a fidelity, for a range of σ, p, and δ. The phase error model was based on 10^{4} normally distributed Monte Carlo samples for each chip configuration, with δ the phase offset standard deviation. Data from each model is compared to the experimentally obtained data, and Bayesian inference learns the likeliest value for each parameter.
Consider a system described by a known model M(σ) with free parameter σ, a set of N observables \({\Pi} = \{ \pi _i\} _{i = 1}^N\) and a data set \(X = \{ x_i\} _{i = 1}^N\): the general aim of Bayesian parameter estimation is to find the parameter \(\bar \sigma\) that best describes the data outputted by the system. Learning \(\bar \sigma\) relies on the estimation of likelihoods, over a discretised space \(\{ \sigma _k\} _{k = 1}^K\) of K possible σ_{k}: \(L(\sigma _k) = \mathop {\prod}\nolimits_{i = 1}^N P(x_i\sigma _k,\pi _i),\) where P(x_{i}σ_{k}, π_{i}) is the probability of observing x_{i} given model parameter σ_{k} and measured the observable π_{i}. This probability can be calculated from the frequency of the observed data x_{i} over many samples of simulated data \(\tilde x_i\). We can therefore derive the probability of σ_{k} being the parameter that best describes the data by applying Bayes’s rule:
thus retrieving a probability distribution for each parameter. We have assumed the measurements to be uncorrelated and the a priori distribution of the parameters P(σ_{k}) to be constant over the discretised range.
Data availability
Data and computer code that support the findings of this study are available at the University of Bristol’s data repository, data.bris (Digital object identifier: 10.5523/bris.2nk9fm85ssqaa2lyu4trhp5rqs). Other information is available from the authors upon reasonable request.
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Acknowledgements
This work was possible with the invaluable help of Damien Bonneau, Chris Sparrow, Mercedes GimenoSergovia, Sam Pallister, Will McCutcheon, Stefano Paesani, Eric Johnston, Laurent Kling, Graham D. Marshall, and John G. Rarity. It was generously supported by EPSRC Programme Grant EP/L024020/1, the EPSRC Quantum Engineering Centre for Doctoral Training EP/L015730/1, and the ERC Starting Grant ERC2014STG 640079. J.W.S. acknowledges the generous support of the Leverhulme Trust, through Leverhulme Early Career Fellowship ECF2018276. MGT acknowledges support from EPSRC Early Career Fellowship EP/K033085/1.
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J.W.S., R.S. and J.C.A. conceived the device. J.C.A. and C.V. designed and carried out the experiments and experimental modelling. J.W.S. and M.G.T. supervised the project. All authors analysed the results and wrote the manuscript.
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Adcock, J.C., Vigliar, C., Santagati, R. et al. Programmable fourphoton graph states on a silicon chip. Nat Commun 10, 3528 (2019). https://doi.org/10.1038/s4146701911489y
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DOI: https://doi.org/10.1038/s4146701911489y
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