Efficient generation of entangled multiphoton graph states from a single atom

The central technological appeal of quantum science resides in exploiting quantum effects, such as entanglement, for a variety of applications, including computing, communication and sensing1. The overarching challenge in these fields is to address, control and protect systems of many qubits against decoherence2. Against this backdrop, optical photons, naturally robust and easy to manipulate, represent ideal qubit carriers. However, the most successful technique so far for creating photonic entanglement3 is inherently probabilistic and, therefore, subject to severe scalability limitations. Here we report the implementation of a deterministic protocol4–6 for the creation of photonic entanglement with a single memory atom in a cavity7. We interleave controlled single-photon emissions with tailored atomic qubit rotations to efficiently grow Greenberger–Horne–Zeilinger (GHZ) states8 of up to 14 photons and linear cluster states9 of up to 12 photons with a fidelity lower bounded by 76(6)% and 56(4)%, respectively. Thanks to a source-to-detection efficiency of 43.18(7)% per photon, we measure these large states about once every minute, which is orders of magnitude faster than in any previous experiment3,10–13. In the future, this rate could be increased even further, the scheme could be extended to two atoms in a cavity14,15 or several sources could be quantum mechanically coupled16, to generate higher-dimensional cluster states17. Overcoming the limitations encountered by probabilistic schemes for photonic entanglement generation, our results may offer a way towards scalable measurement-based quantum computation18,19 and communication20,21.

Entanglement plays a crucial role in quantum information science.For multi-qubit systems, many of the states considered, e.g. for entanglement purification, secret sharing, quantum error correction as well as interferometric measurements, belong to the family of graph states [9].Two prominent examples are Greenberger-Horne-Zeilinger (GHZ) and cluster states, which are central ingredients for various measurementbased quantum information protocols [19][20][21].One-way quantum computing [18], for instance, represents a conceptually appealing alternative to its circuit-based counterpart.Instead of carrying out unitary quantum-logic gates, computation relies on adaptive single-qubit measurements.This operational easing comes at the price that a multi-qubit entangled resource state, a cluster state, needs to be prepared in advance.
Although multi-qubit entanglement has been demonstrated on various platforms [3,[22][23][24][25][26], only small-scale implementations of measurement-based quantum computing (MBQC) have been realized so far [10,27,28].Amongst these platforms, optical photons stand out as qubit carriers as these suffer negligible decoherence and benefit from crosstalkfree single-qubit addressability and measurement capabilities with off-the-shelf components.However, the most common sources for entangled photons are based on spontaneous parametric down conversion (SPDC).This scheme is inherently probabilistic and thus makes scaling up to larger states an increasingly difficult challenge, even for a moderate number of qubits.
To address this issue, deterministic schemes have been proposed [4][5][6].These employ a single-spin memory qubit that mediates entanglement over a string of sequentially emitted photons.This approach is resource efficient as it permits the generation of in principle arbitrarily many entangled photons from a single device.First experiments along these lines have been performed with quantum dots [11,12] demonstrating entanglement of up to three and four qubits, respectively, in a linear cluster state.Low photon generation and collection efficiencies, a noisy semiconductor environment, or the need for a probabilistic entangling gate were amongst the biggest obstacles for reaching higher photon numbers.Recent experiments with Rydberg superatoms [13,29] demonstrated GHZ states of up to six photons.While the single-emitter strategy could in principle provide a stepping stone for photonic quantum computation, no implementation has demonstrated a performance that beats or even compares to the SPDC approach [3].
Here we produce large and high-fidelity photonic graph states of the GHZ and cluster type.Inspired by the proposals of ref. [4][5][6], which we adapt to our physical system, we use a cavity quantum electrodynamics (CQED) platform as an efficient photon source [30][31][32][33][34] and, for the first time, surpass the state-of-the-art SPDC platform.Arbitrary single-qubit rotations between photon emissions allow for the flexible preparation of different types of states in a programmable fashion.We generate and detect GHZ states of up to 14 photons and linear cluster states of up to 12 photons with genuine multipartite entanglement.In principle, higher dimensional cluster states can be created by coupling several sources [17], e.g.via optically mediated controlled NOT gates of the kind demonstrated recently [16].By virtue of this feature, so far unique to the atomic CQED platform, our technique supports modular extension towards scalable architectures for one-way quantum computation [18,19] as depicted in Fig. 1a.

EXPERIMENTAL SETUP AND PROTOCOL
Our apparatus is shown schematically in Fig. 1b.It consists of a single 87 Rb atom at the center of a high-finesse optical cavity.A magnetic bias field oriented parallel to the cavity direction defines the quantization axis and gives rise to a Zeeman splitting with Larmor frequency ω L = 2π • 100 kHz.Each source produces a one-dimensional (1D) photonic cluster state while at the same time entanglement is distributed across the emitters by ancillary photons being successively reflected off each atom-cavity system [16].Thus, a two dimensional fabric of photonic cluster states is woven from the individual 1D chains readily serving as a resource for MBQC.b, Experimental setup.A single 87 Rb atom coupled to a high-finesse cavity emits a stream of entangled photons.Various lasers directed onto the atom from the side allow to control the photon emission and manipulate the atomic state to realize the desired protocol.The chain of photons is detected by a polarization resolving detection setup composed of a polarizing beam splitter (PBS) and two detectors (Det1 and Det2).A fast polarization electro-optic modulator switches between two predefined settings such that each individual photon can be measured in either the Z or X basis.A half-wave plate before the PBS rotates the detection basis from X to Y and arbitrary superpositions thereof.c, Protocol for cluster/GHZ state generation divided into three steps.Initialization of the atom and first photon emission, cycling process where the atomic qubit is rotated by θ = 0 or π/2 and a new entangled photon is emitted, closing of the protocol by emission of a last photon while disentangling the atom from the generated string of photons.
Multiple laser beams propagating perpendicular to the cavity allow for various manipulations such as state preparation by optical pumping and coherent driving of Raman transitions between the hyperfine ground state manifolds with energy selectivity provided by the Zeeman splitting.The cavity serves as an efficient light-matter interface for atom-photon entanglement [7] with an optical cooperativity of C ≈ 1.5 (Methods).A vacuum-stimulated Raman adiabatic passage (vSTI-RAP) enables the generation of photons with high indistinguishability stemming from accurate control over their temporal wavefunction [30].Photons which are outcoupled from the cavity are analyzed with a polarization resolving detection setup mainly consisting of a polarizing beam splitter and a pair of superconducting nanowire single-photon detectors.Additionally, an electro-optic modulator is employed for fast selection of the measurement basis by switching between the Z basis (right and left circular polarization) and the X basis (horizontal and vertical polarization).When set to the X basis, a half-wave plate can optionally be placed in order to rotate the detection basis along the equator of the Bloch sphere.
The experimental protocol for generation of entangled photons in essence consists of a periodic sequence of photon generations interleaved with single-qubit rotations performed on the atom.The sequence is displayed in Fig. 1c including the corresponding processes in the atomic level diagram.We first initialize the atom in the state |F = 2, m F = 0⟩ by optical pumping.Here, we write the atomic state as |F, m F ⟩ where F denotes the total angular momentum and m F its projection along the quantization axis.Then, we apply a control pulse (1.5 µs) which induces the vSTIRAP process generating a photon (300 ns FWHM) entangled in polarization with the atomic state.This process can be written as |2, 0⟩ → (|1, 1⟩ |R⟩ − |1, −1⟩ |L⟩) / √ 2, where |R/L⟩ denotes right/left circular polarization of the photon and {|1, 1⟩ , |1, −1⟩} serves as our atomic qubit basis.We then perform a single qubit rotation R θ of angle θ (1) on the atom.For cluster states θ = π/2, for GHZ states no rotation is performed, i.e. θ = 0. Afterwards, we transfer the qubit from |1, ±1⟩ to |2, ±2⟩ (2).Both steps (1) and ( 2) are realized via a series of Raman pulses with a 790 nm laser (Methods).Finally, we induce the vSTIRAP process (3) by applying a control pulse which produces a photon (4) and takes the atom back to the states |1, ±1⟩.Steps (2-4) can be summarized by writing |1, ±1⟩ → |1, ±1⟩ |R/L⟩.One photon production cycle consisting of the steps (1-4) takes 200 µs (50 µs) for the cluster (GHZ) state sequence.It is repeated N − 2 times, each iteration adding another qubit to a growing chain of entangled photons.For the final photon however (closing), the atomic qubit is transferred from |1, ±1⟩ to |2, ∓1⟩ (instead of |2, ±2⟩) such that in the subsequent emission process the atom ends up in |1, 0⟩, which readily disentangles it from the photonic state.We note that for cluster states initializing as well as disentangling the atom are not strictly necessary as the same can be achieved by appropriate Z basis measurements of the first and last photon [6].In the case of GHZ states however, the protocol must be performed as described in Fig. 1c to obtain an Nphoton state of the form

GREENBERGER-HORNE-ZEILINGER STATES
We start the experiment by producing GHZ states.In contrast to cluster states, GHZ states are more sensitive to noise and require a higher level of control in their preparation process.Regardless, since their density matrix contains only four non-zero entries, it is much easier to measure the fidelity F N of an N -photon GHZ state [35] than for a cluster state, despite the large Hilbert space of dimension 2 N .Therefore, the quantitative analysis of a multi-photon GHZ state, besides representing an interesting result by itself, provides a useful tool for benchmarking and gives insights into the inner dynamics of our system.
For estimation of the fidelity it is sufficient to measure the non-zero elements on the diagonal and off-diagonal of the density matrix separately.The diagonal elements represent the populations P N of the |R⟩ ⊗N and |L⟩ ⊗N components of the state and can be obtained by measuring all photons in the Z basis.The corresponding experimental data, shown in Fig. 2b in blue, agree well with the ideal GHZ state, for which P N = 1, with only a weak dependence on N .In order to demonstrate that the states |R⟩ ⊗N and |L⟩ ⊗N are in a coherent superposition, we set the measurement basis to (|R⟩ ± e iϕ |L⟩)/ √ 2 where ϕ ∈ [0, π], thus spanning the full equator of the Bloch sphere.This allows us to measure the characteristic parity oscillations which behave as cos(N ϕ) (Methods), see Fig. 2a.The coherences C N of the density matrix are extracted from the visibility of the oscillations for all photon numbers up to N = 10.For 14 photons the coincidence rate drops significantly due to the finite photon production efficiency.To acquire enough data we only measure the parity for ϕ = 0 which is indicated by the yellow diamond in Fig. 2b.Eventually, the fidelity is calculated via the formula F N = (P N + C N )/2 and is shown in Fig. 2b in red.As only a single measurement setting was used for C 14 , we additionally provide a lower bound for the fidelity based on an entanglement witness (Methods).With this we prove genuine 14-photon entanglement with a fidelity F 14 ≥ 76(6)%, surpassing the 50% threshold by more than 4 standard deviations.To the best of our knowledge, this is the largest entangled state of photons to this day.
Within the measured range we observe that the decay of P N , C N and F N as a function of photon number is well captured by a linear model with a slope of 0.86(9)%, 1.3(2)% and 1.04(9)% per photon, respectively.By extrapolation of this trend we estimate that the fidelity will cross the 50% threshold at around 44 qubits.The remarkably slow decay in fidelity is particularly astonishing as we observe very little decoherence even when the sequence is deliberately chosen to exceed the intrinsic coherence time of the atomic qubit (∼1 ms).This behaviour is explained by a dynamical decoupling effect built into the protocol, which arises from the opposite signs of the Zeeman splitting in the two hyperfine ground state manifolds.Hence, the qubit precession is reversed every time the atom is transferred from |F = 1⟩ to |F = 2⟩ or vice versa, which can be seen as two spin-echo pulses for every photon production cycle.While this mechanism contributes to the high-visibility fringes seen in Fig. 2a, no extra effort is needed to exploit it (Methods).We currently attribute the main source of infidelity to the vSTIRAP.This can be explained by the finite cooperativity that allows for unwanted paths in the emission process (Methods).

CLUSTER STATES
The characterization of cluster states is more demanding as the density matrix contains many non-zero elements.We therefore use the entanglement witness W proposed in ref. [36], which is based on the stabilizer formalism.A lower bound of the fidelity can be derived from W requiring only two local measurement settings XZXZ... and ZXZX... (Methods).Compared to quantum state tomography, this has the advantage of a tremendous reduction in experimental overhead, but comes at the cost of a potentially significant underestimation of the true state fidelity.Nonetheless, the experimental results displayed in Fig. 3a exceed the 50% threshold for all measured points.Here, the data only includes events in which exactly N photons are detected for a sequence of N consecutive generation attempts.For the largest cluster state of 12 photons we find the fidelity to be lower bounded by 56(4)%.
Comparing the results to the GHZ states in Fig. 2 we notice a significantly faster decay of the fidelity (3.6(2)% per photon).
Besides the large number of Raman transfers in the protocol (5 transfers per cycle, see Methods), we attribute this mainly to the lower bound which by construction underestimates the fidelity.A tighter lower bound that was recently formulated [37] could provide a higher fidelity estimate in future experiments.
In addition to providing a lower bound for the fidelity, we now present the measured stabilizer operators defined as S k = Z k−1 X k Z k+1 (Fig. 3b).Here k ∈ {1, 2, ..., N }, Z 0 = Z N +1 = 1, and X k and Z k denote the respective Pauli matrices acting on the k th qubit.In this scenario events in which three consecutive photons, k − 1, k and k + 1, are detected in the appropriate basis contribute to the stabilizer S k .In principle arbitrarily many stabilizers could be measured by Measured N -photon coincidence rate.The data (blue for GHZ and red for cluster states) represent the number of coincidences divided by the total measurement time.From an exponential fit to the data we extract a single photon generation and detection probability of η = 43.18(7)%.The light colored lines represent the estimated coincidence rate assuming a loss-corrected efficiency of η = 0.66 (see also ref. [30]).Equivalent rates are plotted for state-of-the-art SPDC [3], quantum dot (QD) in purple [11] and orange [12], and Rydberg-based [13] (Ryd) systems.Error bars are smaller than the markers.
repeating the protocol for a corresponding number of cycles.
Here however, we terminate the sequence at k = 15.We find an average of ⟨S 1 ⟩ = 96.13(9)%and ⟨S k ⟩ = 92(1)% for k ≥ 2, indicating a large overlap of the generated state with the target linear cluster state.

COINCIDENCE RATE
We emphasize that the ability of producing entanglement of up to 14 photons is based, on the one hand, on the excellent coherence properties of the atom, and on the other hand, the large photon generation and detection efficiencies.The latter is crucial as the success probability p s of detecting a coincidence of N consecutive photons scales exponentially with the photon number, p s = η N .Here, η denotes the probability to generate and detect a single photon for a given attempt.We can express η as the product of the source efficiency η 0 , i.e. the probability of producing a photon at the output of the cavity, and the detection efficiency η d .It is clear that a low efficiency η ≪ 1 poses a great obstacle to achieving large photonic states within reasonable measurement times.
Fig. 4 shows the raw rate of multi-photon coincidences as a function of photon number N including post-selection and experimental duty cycle.The experimental sequence consists of 14 (12) consecutive photon generation attempts with all timing parameters identical to the GHZ (cluster) protocol and a new run starting every 1.1 ms (3.0 ms).The shown data (blue for GHZ and red for cluster states) is the coincidence count rate of events in which N consecutive photons were detected starting from the first attempt.For instance, for the largest state of 14 photons we recorded 151 coincidences in 7 hours of experimental runtime, equivalent to roughly one event every three minutes.From an exponential fit to the data we extract the overall single-photon generation and detection efficiency η = 43.18(7)%.We estimate the intrinsic generation efficiency η 0 to be 66% mainly limited by the cooperativity and the escape efficiency (see ref. [30]).Both can be optimized by higher-quality mirrors and a smaller cavity-mode volume.The detection efficiency of η d = 0.7 captures all the remaining loss contributions such as optical elements and detectors.These include free-space-to-fiber couplings (94% twice), propagation through optical fiber (97%), free-space optics (90%) and detector efficiency (90%).Correcting for the detection efficiency η d , we infer an N -photon coincidence rate at the output of the cavity as given by the light blue line in Fig. 4.This represents the limit of our system with the current parameters.As a comparison, we also show the rate of the best available SPDC system as well as deterministic sources based on single quantum dots or Rydberg-blockaded atomic ensembles.Although the repetition rate for these systems is typically many orders of magnitude higher than in our protocol, our system outperforms previous implementations by far in terms of real time coincidence count rate as well as efficiency scaling.

SUMMARY AND OUTLOOK
To conclude, we have presented a scalable and freelyprogrammable source of entangled photons, demonstrating, to our knowledge, the largest entangled states of optical photons to this day.It is deterministic in the sense that no probabilistic entangling gates are required.This gives us a clear scaling advantage over previous schemes.Moreover, the ability to perform arbitrary single-qubit rotations on the emitter provides the flexibility to grow graph states of different types.
At this stage, our system faces mostly technical limitations, such as optical losses, finite cooperativity and imperfect Raman pulses.Even modest improvements in these respects would put us within reach of loss and fault tolerance thresholds for quantum error correction [19,[38][39][40].Hence, a clear path towards one-way quantum computing architectures would be the generation of two-dimensional cluster states by entangling multiple photon sources [17].For example, in a next step two of our systems could be coupled via remote quantum logic gates [16] to produce 2 × N 'ladder' cluster states.Alternatively, entangling operations such as gates or Bell-state measurements could be performed on two (or more) individual atoms as single emitters in the same cavity [14,15].Similar strategies apply for the generation of tree graph states and one-way quantum repeaters [20,21].The present work thus opens up a new road for photonic quantum computation and communication.

Experimental setup
The central component of the setup used in this work is a high-finesse optical cavity with a 87 Rb atom trapped at its center.The cavity consists of two highly reflective mirrors oriented parallel to each other at a distance of 500 µm with an optical mode waist of w 0 = 30 µm.The two mirrors have a transmitivity of T 1 = 100 ppm and T 2 = 4 ppm giving rise to a finesse of F ≈ 60, 000 such that photons populating the cavity mode are outcoupled predominantly through the low-reflective side.The cavity is tuned to the atomic D2 line with a detuning of ∆ c = −150 MHz with respect to the transition |F = 1⟩ ↔ |F ′ = 1⟩.The combined system of the atom and cavity is best described in the framework of cavity quantum electrodynamics with parameters (g, κ, γ) = 2π • (c ge • 10.8, 2.7, 3.0)MHz , g being the atom-cavity coupling strength for the relevant transition, κ the decay rate of the cavity field and γ the free-space atomic decay rate associated with the D2 transition of 87 Rb. c ge is the Clebsch-Gordan coefficient between the relevant excited state (|e⟩) coupling to the vSTIRAP control pulse and the final state (|g⟩) of the photon production process.The above parameters put our system in the intermediate to strong coupling regime with a cooperativity parameter defined as C = g 2 / (2κγ).Note that the specific value of C depends on the transition path associated to a certain excited state.For example, for the emission from |2, ±2⟩ as in the cycling step of the protocol we have |g⟩ = |F = 1, m F = ±1⟩ and |e⟩ = |F ′ = 2, m ′ F = ±2⟩.Hence, we get c ge = 1/4, giving C = 1.8.

Atom loading
Atoms are transferred from a magneto-optical trap (MOT) to the center of the cavity where they are trapped by a twodimensional optical lattice composed of two standing wave potentials, one at 772 nm oriented along the cavity axis and one at 1064 nm propagating perpendicular to the cavity axis.An EMCCD camera detects the atomic fluorescence which is collected via a high NA objective.A single atom is prepared quasi-deterministically by removing any excess atom with a resonant laser beam steered onto the atom via an acousto-optic deflector (AOD).The position of the atom is monitored during the experiment and controlled via appropriate feedback to the optical trapping potential.

Experimental duty cycle and post-selection
Because the atoms have a finite lifetime in the dipole trap, they have to be reloaded regularly.The average trapping time depends significantly on the type of conducted experiment (i.e.heating/cooling mechanisms).For our experiments, we achieved an average trapping time of roughly 20 seconds.The time required for loading and repositioning of the atom after occasional jumps to a different location reduces the experimental duty cycle.By counting the number of experimental runs carried out at a given repetition rate over a longer measurement interval, we evaluate the overall duty-cycle to be close to 50%.
Once a camera image shows that the atom has moved away from the target position, the corresponding data to that image is discarded via post-selection processing.The same applies to images with more than one atom near the cavity center.
Further post-selection is performed by processing the data collected by the single-photon detectors: an experimental run is considered successful when N photons were detected in a row, each within predefined time windows (1 µs width in this work).Note that Fig. 4 shows the coincidence rate after applying post-selection.

Protocol
The full experimental sequence including timings of the optical pulses is shown in Extended Data Fig. 1.As described in the main text, it mainly consists of a repeating sequence of single-qubit rotations and photon emissions (cycling) with additional initialization and closing steps at the beginning and the end.The atom is initialized in the state |2, 0⟩ by optical pumping (5 µs).A square-shaped control pulse (1.5 µs) produces the first photon, thus generating the atom-photon entangled state |1, 1⟩ |R 1 ⟩ − |1, −1⟩ |L 1 ⟩ (up to normalization) where the index '1' refers to the first photon.If no photon was detected, we immediately go back to the state preparation step and another photon attempt.We choose a maximum of seven attempts for the first photon in order to avoid excessive heating of the atom.After a successful first photon detection we start the cycling stage with the single qubit gate, which for cluster states consists of a π/2 rotation contained in three Raman manipulations.First, the population in |1, 1⟩ is transferred to |2, 0⟩ with a π pulse taking 53 µs.Then a π/2 pulse is applied to the transition |1, −1⟩ ↔ |2, 0⟩ realizing the qubit rotation.Afterwards, the population in |2, 0⟩ is transferred back to |1, 1⟩ with another π pulse.The above described operation transforms the basis states as follows: |1, 1⟩ → |1, 1⟩+|1, −1⟩ and |1, −1⟩ → − |1, 1⟩+|1, −1⟩.The whole pulse sequence for the single-qubit gate takes 132.5 µs.For GHZ states the required rotation angle is θ = 0, which means that the qubit rotation can be skipped entirely.In order to produce the next photon we transfer the population from |1, ±1⟩ to |2, ±2⟩ via two sequential Raman π-pulses (790 nm) each taking 21 µs.We then apply a vSTIRAP control pulse leading to a photon emission.The atom-photonphoton state then reads for cluster states, i.e. θ = π/2.The index '2' now refers to the second photon.The cycling step is repeated as many times as desired.In the very last cycle, the closing step is performed.Here, following the qubit rotation the atomic population is transferred from |1, ±1⟩ to |2, ∓1⟩ instead of |2, ±2⟩, which takes 55 µs.Thus, the atom is disentangled in the subsequent photon emission.This step can be seen as an atom-to-photon state transfer, as the atomic qubit is mapped to the polarization state.After the last photon we run a calibration sequence for actively stabilizing the optical power of the laser pulses.Finally, the atom is laser-cooled for several hundred microseconds.The length of a full period of the experiment including calibration and cooling depends on the type of state produced and the number of photons N .It can be as short as 400 µs and as long as 3 ms.

Raman manipulations
The Raman transitions shown as orange and green arrows in Extended Data Fig. 1 are performed with a 790 nm laser.The duration of these transitions make up the most part of the experimental sequence.In principle choosing a higher Rabi frequency could drastically increase the repetition rate of the protocol, but would lead to more crosstalk between the transitions as they would start to overlap in frequency space.As a consequence a compromise between repetition rate and high-fidelity Raman manipulations has to be found.For our choice of experimental parameters we estimate the infidelity per single-qubit rotation to be smaller than 1%.
The Raman transfer in the closing step from |F = 1⟩ to |F = 2⟩ is realized with a 795 nm Raman laser close to the D1-line of Rubidium.For this specific Raman transition we cannot choose a large detuning since this would lead to a destructive interference due to the Clebsch-Gordan coefficients.As a consequence we have a chance of about 5% of spontaneous scattering, which reduces the fidelity.As mentioned in the main text, alternatively the atom can also be disentangled from the photonic state by measuring the most recently generated photon in the Z basis.While this would slightly increase the fidelity, the rate would drop as the detection of an additional photon is required.

Estimation of errors
For GHZ states we observe a total error rate of about 1% per photon.We attribute most of the infidelity to spontaneous scattering during the photon production process, as the vSTI-RAP control pulse couples to the F ′ = 3 excited state of the D2.This opens a decay channel which competes with the coherent emission of the photon.By post-selecting on early photon arrival one can partly filter out events in which scattering has occurred (Extended Data Fig. 4).In the future, this could be eliminated by generating the photons on the D1 line, where no F ′ = 3 state is present.This should significantly improve the error rate.
The same error mechanism applies in the case of cluster states.Additionally, the single-qubit gate implemented with Raman lasers introduces errors, which we estimate to be smaller than 1%.These could be explained by finite frequency resolution, pulse intensity fluctuations as well as drifts in optical alignment.Increasing the Zeeman splitting for instance would be a way to further optimize this process.
Minor sources of error include polarization alignment.For setting the polarization detection basis we use a reference polarizer in front of the cavity and measure the polarization extinction to be on the order of 10,000:1.For switching the detection basis we use a polarization electro-optic modulator (EOM, QUBIG PC3R-NIR) with a switching time of 5 ns.The extinction ratio is specified as >1000:1, whereas we measured values of around 5000:1.
The error rate for cluster states of 3.6% as given in the main text is presumably overestimated due to the definition of the fidelity lower bound.Taking into account the error sources identified above, we estimate the true error rate to be smaller than 2%.With the suggested improvements we expect a reduction well below 1% to be realistic.

Generation efficiency
The intrinsic source efficiency, i.e. the probability of obtaining a photon at the output of the cavity, is given by where C ≈ 1.5 is the cooperativity and η esc ≈ 0.88 denotes the escape efficiency, i.e. the probability of a photon being outcoupled from the output port [30].Note that the above formula is only valid in the case of a single excited state, whereas the efficiency becomes a function of the detuning, η 0 (∆), when multiple excited states are present.
The source efficiency could hence be improved by increasing both the cooperativity and the escape efficiency.As the two parameters are generally not independent, let us assume for simplicity that we reduce the waist of the cavity mode by a factor of 2. This increases the cooperativity by a factor of 4 without altering the escape efficiency.We would thereby improve the source efficiency from 66% to 81%.Furthermore, the efficiency of the detection setup could be improved.For instance, by redesigning and optimization of the setup one could replace a fiber-to-fiber coupling with a fiber splice, eliminate a free-space-to-fiber coupling and reduce the losses from optical surfaces.In this scenario an improvement of the detection efficiency η d from 0.7 to 0.85 seems feasible.Given these realistic improvements the combined source-to-detection efficiency η would reach the mark of 2/3, an important threshold for linear optical quantum computation [40].

GHZ state fidelity
In the mathematical formalism of spin 1/2 particles a GHZ state looks like where in the photonic case |↑⟩/|↓⟩ corresponds to |R⟩/|L⟩.For measuring the diagonal elements of the density matrix, i.e. the populations P N of the |↑⟩ ⊗N and |↓⟩ ⊗N components it suffices to measure all particles in the Z basis to obtain For the coherences we introduce the parity operator [3,35] M ϕ = 0 e −iϕ e iϕ 0 ⊗N describing a measurement of all N particles in the basis |↑⟩ ± e iϕ |↓⟩ / √ 2. Varying the parameter ϕ from 0 to π corresponds to a continuous rotation of the measurement basis along the equator of the Bloch sphere.In the experiment this is achieved by scanning the angle of a half-wave plate in front of the PBS in the detection setup.It is straightforward to show that the expectation value of M ϕ for the ideal GHZ state is These characteristic parity oscillations are what can be seen in Fig. 2a of the main text.The amplitude of the oscillations as obtained from a cosine fit are a measure for the coherences of the density matrix.The fidelity is then obtained from the formula For the largest photon number of N = 14 we chose to measure an entanglement witness derived in ref. [36] in order to obtain a fidelity lower bound.The witness is based on the stabilizer formalism, the stabilizing operators for GHZ states being where k = 1, 2, ..., N and Z k , X k are the Pauli matrices acting on the kth qubit.With this the fidelity is lower bounded by Witnessing cluster states entanglement A lower bound for the fidelity can be derived in a similar fashion for 1D cluster states [36].With the set of stabilizers S k as defined in the main text the bound is given by the inequality It is easy to verify by direct calculation that the terms for even and odd k in Eq. 10 correspond to the local measurement settings ZXZX... and XZXZ..., respectively.As an example, for a four qubit linear cluster state we have Coherence and dynamical decoupling In the main text we already introduced that our system benefits from a built-in dynamical decoupling mechanism due to the level structure of the atomic hyperfine ground states.A measurement of the intrinsic coherence time of the atom can be seen in Extended Data Fig. 3a.Here we look at the overlap between two photons both emitted from the atom with a variable time delay.The first photon is measured in the linear basis (|H⟩ / |V ⟩) which projects the atom onto a superposition of the qubit states |1, +1⟩ and |1, −1⟩.The atomic state then precesses with twice the Lamor frequency.After a certain time t the atomic qubit is read out by mapping it onto a photon which is then measured in the same basis as the first photon.The fidelity, which we define as the projection of the second photon on the first, shows oscillations damped by noise such as magnetic field fluctuations.After roughly 1.2 ms the envelope of the oscillations cross the classical threshold of 0.66 which defines the intrinsic coherence time of the atomic qubit.For the GHZ sequence however, we observe that the effect of decoherence is intrinsically reduced.We can show this by artificially extending the length of the sequence to 1.25 ms for a 6 photon GHZ state.In this case every photon production cycle takes 300 µs.The ratio of time the qubit resides in |F = 1⟩ and |F = 2⟩ can then be varied by scanning the delay τ between the hyperfine transfer from |1, ±1⟩ to |2, ±2⟩ and the vSTIRAP control pulse as illustrated in Extended Data Fig. 3b.For different values of τ we record the parity oscillations similar to Fig. 2a and infer the visibility.From the measured data we can see a clear dependence of the visibility as a function of τ with a rephasing appearing at around 85 µs.The maximum value is roughly equal to the 6 photon coherence displayed in Fig. 2 of the main text (shown as a dashed line for reference), for which the sequence length was only 250 µs.This is strong evidence that a large part of the decoherence is mitigated as an inherent feature of the protocol.

Relative efficiency
Extended Data Fig. 4. Infidelity induced by vSTIRAP process.Two photons are generated in subsequent cycles of the GHZ protocol and measured in the R/L basis.Their correlation (red) is analyzed as a function of maximum permitted arrival time tmax with respect to the beginning of the emission process.The relative efficiency (blue) displays the number of counts detected up to tmax as opposed to the full photonic wave packet.The correlation decreases as a function of tmax, which we attribute to spontaneous scattering events induced by the vSTIRAP control pulse.The dashed line marks the value of tmax used in this work.

Figure 2 .
Figure 2. GHZ states.a, Parity measured in the basis (|R⟩ ± e iϕ |L⟩)/ √ 2 for photon numbers N = 2, 6, 10 (see Extended Data Fig. 2 for the full data set).The coherences CN are derived from the visibility of the oscillations.b, Populations PN (blue squares), coherences CN (yellow circles) and extracted fidelities FN (red triangles) with weighted linear fits as solid lines.The fidelity is calculated via the formula FN = (PN + CN )/2.For N = 14 the yellow diamond indicates that the coherence was derived from a single measurement setting with ϕ = 0.The grey dashed line marks the classical threshold of 0.5.Error bars represent 1 standard deviation.

Figure 3 .
Figure 3. Cluster states.a, Lower bound of the fidelity as provided by ref.[36] obtained from two measurement settings.For the measured data the lower bound exceeds the threshold of 50% indicated by the grey dashed line.b, Measured stabilizers S k given by the expectation value ⟨Z k−1 X k Z k+1 ⟩ up to k = 15.All measured stabilizers are larger than 0.9, marked by the dashed line.Error bars represent 1 standard deviation.