Fusion of deterministically generated photonic graph states – Nature

Experimental setup

The apparatus used in our work consists of a single-sided high-finesse cavity in which we optically trap two rubidium atoms. Most experimental details about the setup including the cavity quantum electrodynamics parameters have already been described elsewhere¹⁶. In the following, we provide further information that is important for the current work.

The atoms are trapped in a two-dimensional optical standing-wave potential formed by two pairs of counter-propagating laser beams. The first is a retro-reflected laser at a wavelength of λ = 1,064 nm along the x axis. The second propagates inside the cavity mode along the y axis with λ = 772 nm. The atoms are loaded from a magneto optical trap to the cavity centre using a second 1,064-nm running-wave laser. The light scattered by the atom during laser cooling is imaged by means of the objective onto an electron-multiplying charge-coupled device camera to spatially resolve the position of the atoms. After each loading attempt, we find a random number of atoms n at random positions. The experimental control software identifies atom pairs with a suitable relative distance d. If no such atom pair is present, a new loading attempt starts immediately. Otherwise, a tightly focused resonant laser beam, propagating through the objective and steered by the AOD, removes the n − 2 unwanted atoms. The x component of the centre-of-mass position of the atom pair (x₂ + x₁)/2 is then actively stabilized to the centre of the cavity mode by acting on the relative phase of the 1,064-nm counter-propagating laser beams. The y components y₁ and y₂ are controlled by modulating the optical power of the 772-nm intra-cavity trap until the atoms are found in a desired position.

Fusion gate and post-selection

For a fusion gate to be successful, two photons have to be detected, as described in the main text. Mathematically, this can be understood by considering two atom–photon entangled states of the form

The relative minus sign in the above equation arises from the Clebsch–Gordan coefficients in the two emission paths. Applying the projector ⟨R|⟨L| to the product state |ψ_AP⟩ ⊗ |ψ_AP⟩ corresponds to the detection of an R and an L photon, signalling a successful fusion. This leaves us with the |Ψ⁺⟩ Bell state. Here we implicitly assumed that the two photons occupy the same spatiotemporal mode function. In the experiment, however, their temporal wave packet may not be perfectly indistinguishable, leading to an incomplete erasure of which-path information. Such imperfection can arise from spontaneous scattering by means of the excited state or from unbalanced atom–cavity or atom–laser coupling. This effect becomes visible when post-selecting on the arrival time of the photons. The influence of the arrival time on the fidelity of the atom–atom Bell state is summarized in Extended Data Fig. 1. Panel a shows the intensity profile of the photon temporal wave function as a function of t_R,L, with t_R and t_L being the arrival times of the R-polarized and L-polarized photons produced in the fusion process, respectively. Events in which a photon arrives outside the time interval marked by the dashed lines are discarded. This interval contains about 98% of all single-photon counts. Panel b is a two-dimensional density plot of the number of two-photon events versus arrival times t_R and t_L. We can see that most events lie in the vicinity of the point t_R = t_L = 200 ns. The dashed line encloses the region defining the post-selection criteria, which we specify in more detail below. Panel c is a density plot similar to b showing the fidelity as a function of t_R and t_L. We find that the fidelity is highest near the diagonal of the plot, that is t_R ≈ t_L. This motivates our choice of the post-selection region enclosed by the dashed line. Pixels for which we did not acquire enough data to compute the fidelity are shown in white. The fidelity is computed using the formula

Here XX, YY and ZZ are two-qubit operators consisting of the respective Pauli operators. In panels d–f, we analyse their expectation values ⟨XX⟩, ⟨YY⟩ and ⟨ZZ⟩ as a function of arrival time difference |t_R − t_L|. We plot the expectation value both for |t_R − t_L| = τ (orange) and |t_R − t_L| ≤ τ (purple), that is, the cumulative expectation value. We find all correlators to be in good agreement with the ideal case, for which we expect ⟨XX⟩ = ⟨YY⟩ = 1 and ⟨ZZ⟩ = −1. The high fidelity of the two-atom Bell state is also an indicator of a high photon indistinguishability. The dashed lines indicate the maximum value of τ, that is, |t_R − t_L| ≤ τ, chosen for the data presented in Fig. 1c.

Post-selection criteria

For the data in Extended Data Fig. 1c, as well as the data presented in the main text, we apply two post-selection steps. The first step consists of restricting the absolute detection time of the photons to a predefined interval of 1 μs width (see dashed lines in Extended Data Fig. 1a). This step applies to both single-photon and two-photon events. The second post-selection condition involves the relative arrival time difference |t_R − t_L| in the case of two-photon events and therefore only applies to photons generated in the fusion process. The diagonal dashed lines in Extended Data Fig. 1b,c mark the condition |t_R − t_L| ≤ τ_max = 250 ns. Events in which the photons are detected with a relative delay larger than τ_max are discarded. In about 80% of experimental runs, the two photons fall within the interval of τ_max.

As stated in the main text, the atom–atom Bell-state fidelity ranges between 0.851(6) and 0.963(8). The first number refers to the scenario in which no post-selection on the photon arrival time is applied. The second number is obtained when restricting the photon arrival times to t_R,L ≤ 500 ns and |t_R − t_L| ≤ 20 ns. In this case, the post-selection ratio is about 15%.

The above numbers refer to the scenario in which the atom is initialized to |F = 2, m_F = 0⟩ before photon generation. However, in the ring and tree states protocol, the last fusion step consists of a two-photon emission from |F = 2, m_F = ±2⟩. In this case, the photon wave packet is slightly longer, as the m_F = ±2 Zeeman sublevels couple to different excited states in the emission process. Here we apply the same 1-μs time interval as for the m_F = 0 case, as at least 95% of the photon wave packet is enclosed by this window. However, for the two-photon events in the fusion process, we choose a maximum time difference of τ_max = 400 ns to accommodate for a post-selection fraction of about 80%, similar to the m_F = 0 case.

Atom readout

At the end of the generation sequence for tree and ring graph states, the atomic qubits are still entangled with the photons previously generated. One way to measure the atomic qubits is to perform an atom-to-photon state transfer, as done in ref. ¹⁶. Here the qubit is mapped from |F = 1, m_F = ±1⟩ to |F = 2, m_F = ±1⟩ before photon production. In this way, the qubit is fully transferred to the photon, which can then be measured optically. In this work, however, we chose another technique to measure the atomic qubit. For a Z measurement, we transfer the qubit to |F = 2, m_F = ±2⟩ and generate a photon measuring it in the R/L basis. Detecting an R (L) photon projects the atomic qubit onto the state |0⟩_S (|1⟩_S). When measuring the qubit in X or Y, we set the basis directly on the atomic qubit with a π/2 pulse whose phase is tuned according to the basis. The advantage of this scheme is that it can be repeated until success in the case of photon loss, thus increasing the overall efficiency of the state readout. However, as errors are more likely to occur after many repetitions, we limit the number of attempts to three.

Detailed protocol description

In the following, we will describe the generation protocol for the ring and tree graph states with explicit expressions for each step. In the derivation, we do not explicitly include the free evolution of the atomic qubit. In the experiment, the phases that arise from the qubit oscillation are tracked by measuring the stabilizer operators as a function of certain timing parameters related to, for instance, Raman transfers and photon emissions. Notably, these phases may be tuned for each atom independently by varying the respective time of the photon-production pulse.

Ring states

We first describe the protocol of the ring graph states and choose the pentagon ring as a specific example. The box-shaped and hexagon-shaped graphs are obtained from a similar protocol, only omitting a single π/4 rotation. A sketch of the experimental sequence is given in Extended Data Fig. 2a.

The first step of the protocol is to entangle the two atoms and prepare them in the Bell state (|{Psi }^{+}rangle =({|01rangle }_{{rm{S}}}+{|10rangle }_{{rm{S}}})/sqrt{2}). To obtain the pentagon graph, which has an odd number of vertices, we need to apply a global −π/4 pulse. This ‘pushes’ the two qubits apart, forming two separate vertices with an edge between them (Extended Data Fig. 2a, (2)). The corresponding state (omitting normalization constants) reads

Here we have substituted the transformations

and used θ = −π/4, as well as (cos left(frac{{rm{pi }}}{8}right)=frac{sqrt{2+sqrt{2}}}{2}) and (sin left(frac{{rm{pi }}}{8}right)=frac{sqrt{2-sqrt{2}}}{2}). Subsequently, each atom emits a photon, giving

followed by a π/2 rotation on the atomic qubits:

which is equal to a four-qubit linear cluster state with the atoms at both ends of the chain. Note that the (global) π/2 pulse affects only the spin component of the multi-qubit state. We perform another photon production on both spins and obtain

We apply a Z gate to qubit 6 and a Hadamard to qubits 1 and 6 (indices run from left to right).

Now we perform the fusion operation on qubits 1 and 6 and obtain

Here we have moved the second spin qubit to the first position and reintroduced the logical qubit encoding using |0⟩_L and |1⟩_L. Furthermore, we have added a normalization constant. The above expression represents the state that corresponds to the graph shown in Extended Data Fig. 2c. The measured stabilizer expectation values are shown in Extended Data Fig. 2b.

Tree states

We now describe the experimental protocol for generating the target state of the form

We start by preparing both atoms in the |F = 2, m_F = 0⟩ state, followed by three sequential photon-production cycles on each atom in parallel. From this, we obtain the tensor product of two GHZ states, each consisting of one atom and three photons (see also ref. ¹⁶). Omitting normalization constants, we can write the state as

Note that the second term carries a relative minus sign with respect to the first term. This is reflected in the parity measurement shown in Fig. 3b. We now perform a Hadamard gate on all qubits except qubits 2 and 6 (indices run from left to right) and obtain

For the atoms, the Hadamard is carried out with a Raman laser (see Fig. 1e), whereas for the photons, it is absorbed into the setting of the measurement basis.

We now merge both branches into one larger graph state by applying the fusion gate. Hence we generate two photons from the atoms with the global STIRAP control laser. Detecting one photon in R and one in L effectively projects the atoms onto the subspace {|01⟩_S, |10⟩_S}.

For convenience, we have moved the second spin qubit to the first position in the above expression, which allows us to express the two atoms as a logical qubit encoded in the basis {|0⟩_L ≡ |10⟩_S, |1⟩_L ≡ |01⟩_S}. Adding a normalization constant, we can then write the final state as

This is equal to the expression in equation (12), with the only difference being that the root qubit is now redundantly encoded by the two atoms. Alternatively, it would be possible to remove one of the atoms from the state by an X basis measurement.

Coincidence rate

For each multi-qubit state, the typical generation and detection rate is between 0.4 and 2.3 coincidences per minute. The total number of events as well as the total measurement time are summarized in Extended Data Table 1 for each graph state generated in this work. These numbers include all post-selection steps as described above.

Entanglement witness and fidelity bounds

To quantify the agreement between the experimentally produced multi-photon state and the target state, we use an entanglement witness. This has the advantage that we can derive a lower bound of the fidelity without the need for full quantum-state tomography. The fidelity of a density matrix ρ with respect to the target state |ψ⟩ is defined as

Using the stabilizers, we can express the projector to the target state as

Here we have written the projector as a product of two terms G_a and G_b associated with two sets of stabilizers a and b. Each set a/b can be measured with a single local measurement setting M_a/M_b. These only involve measurements in the X or Z basis for every qubit. We can then write the projector in terms of G_a and G_b, giving

As the stabilizers S_i take the values +1 or −1, the product terms G_a and G_b are either 1 or 0. We conclude that (1 − G_a)(1 − G_b) is non-negative. Omitting this term, we find the lower bound

The above expression is applicable if the stabilizers can be divided into two sets a and b, each of which can be measured with a single measurement setting (M_a and M_b). In the context of our experiment, this applies to tree graph states as well as ring graph states of even parity, that is, an even number of vertices. To the best of our knowledge, there is no equivalent method for ring graph states of odd parity, such as the pentagon graph, and a fidelity lower bound cannot be derived.

We can further derive a fidelity upper bound based on the terms G_a and G_b. First, for any pure state |ψ⟩, we have

by direct application of the Cauchy–Schwarz inequality. The terms (1 + S_i)/2 are projectors, because ({S}_{i}^{2}=1) and therefore

By construction, the stabilizers S_i commute and therefore the projectors (1 + S_i)/2 commute as well. Hence, because G_a/b are products of commuting projectors, G_a and G_b themselves are also projectors:

Equation (21) can then be simplified as (langle psi | {G}_{a}{G}_{b}| psi rangle le sqrt{langle psi | {G}_{a}| psi rangle langle psi | {G}_{b}| psi rangle }).

Then, to generalize to mixed states, we write the mixed state ρ as a linear combination of pure states, that is, (rho ={sum }_{k}{p}_{k}left|{psi }_{k}rightrangle leftlangle {psi }_{k}right|), and apply the above inequality to each of them:

We identify the right term as a scalar product of two vectors and again use the Cauchy–Schwarz inequality

which shows the upper bound of the fidelity

In the next section, we will use both fidelity bounds for a comparison between the experimental data and the expected fidelity.

Estimation of errors

In our previous work¹⁶, we identified some error mechanisms present in our system. For single-emitter protocols, the main error sources are spontaneous scattering in the photon-emission process (about 1% per photon) and imperfect Raman rotations (about 1% per π/2 pulse). In the following, we discuss several more mechanisms that could negatively affect the fidelity. In some cases, the effect of these mechanisms on the fidelity of multi-qubit entangled states is difficult to quantify because of the complexity of the entanglement topology and the protocols to generate it. Furthermore, measuring the fidelity of multi-qubit states is a non-trivial task and our measurement setup only allows us to extract a lower and an upper bound of the fidelity.

Fusion gate

For the two-emitter protocols developed in this work, the cavity-assisted fusion gate is probably the largest source of error. As shown in the main text, this mechanism can be used to prepare the |Ψ⁺⟩ Bell state with a fidelity ranging between 0.85 and 0.96, depending on how strictly we post-select on the arrival time of the photons. The fact that the fidelity decreases with a larger arrival time difference τ (see Extended Data Fig. 1) can be explained by an imperfect indistinguishability of the photons involved in the fusion process. For the standard value of τ_max = 250 ns, the fidelity of the |Ψ⁺⟩ Bell state is 0.92. This number includes state readout of the two atoms, each of which is expected to introduce an error similar to a photon emission (roughly 1%). We conclude that the infidelity from the fusion process is on the order of 6%.

Decoherence

Another potential source of infidelity is atomic decoherence caused by magnetic-field noise or intensity fluctuations of the optical-trapping beams. We have measured the coherence time of the atomic qubit T₂ to be approximately 1 ms. However, the atomic qubit is largely protected by a dynamical decoupling mechanism that is built into the protocol¹⁶, thereby extending the coherence time. The exact extent to which this mechanism takes effect depends on the specific timing parameters in the sequence and the frequency range in which the noise sources are most dominant (for example, magnetic-field fluctuations). Therefore, it is difficult to quantify how much the decoherence translates into infidelity of the final graph state. Furthermore, different types of graph state are more or less susceptible to noise⁴⁴. It is therefore not straightforward to theoretically model the role of decoherence in the fidelity of the final multi-partite entangled state.

Qubit leakage

During the protocol, the emitter qubits are continuously transferred between different atomic states. These states are |1, ±1⟩, |2, ±2⟩ and |2, 0⟩, in which we again write the state as |F, m_F⟩ with the quantum numbers F and m_F. However, there seems to be a low probability that, during the emission process, the atom undergoes a transition to |1, 0⟩ (instead of |1, ±1⟩). This is readily explained by and consistent with the finding of spontaneous scattering during the vSTIRAP process, but may equally result from a contamination of σ⁺/σ⁻ polarization components in the vSTIRAP control laser. The latter is in turn caused by either an imperfect polarization setting or longitudinal polarization components owing to the tight focus of the beam. The unwanted σ⁺/σ⁻ components couple to the (left|{F}^{{prime} }=1,{m}_{F}^{{prime} }=pm 1rightrangle ) states and can thus drive a two-photon transition to |F = 1, m_F = 0⟩. This process results in the atom leaving the qubit subspace but, unfortunately, such an event remains undetected. If the protocol resumes with a Raman π/2 pulse, the parasitic population in |1, 0⟩ is then partly transferred to |2, ±1⟩, as the corresponding transitions have the same resonance frequency. A subsequently emitted photon will then yield a random measurement outcome, which is detrimental to the fidelity of the state.

The leakage mechanism described above is difficult to quantify, mainly because our experiment lacks an m_F-selective state readout. We do however estimate that the longitudinal polarization components of the addressing beam have a relative amplitude on the order of about 1%, contributing to each single-photon emission. For the global beam, this effect is negligible owing to a larger focus.

Other sources of error

Other sources of error include drifts of the optical fibres, such as for the Raman beam, the global and addressing vSTIRAP beam or the optical traps, as well as the magnetic field. Furthermore, the position of the atoms is not fixed but varies from one loading attempt to another. In this work, we chose position criteria that are less strict than those in ref. ¹⁶, to increase the data rate. In combination with the drifts mentioned above, this leads to a variance in coupling between the atoms and the cavity, as well as the atoms and different laser beams. As a consequence, this may affect the fidelity of different processes, such as the fusion gate or Raman transfers. Furthermore, a drift of the magnetic field or the light shift induced by the optical trap can influence the phase of the atomic qubits at different stages of the protocol.

A way to reduce the overall infidelity would be to increase the cooperativity C. This would reduce the effect of spontaneous scattering, improve photon indistinguishability and thereby increase the fidelity of the fusion process and partly mitigate the qubit-leakage error. Photon emission through the D₁ line of rubidium would have a similar effect, owing to a larger hyperfine splitting in the 5²P_1/2 excited state. Another strategy to improve the system would be a better control of the atom positions by using more advanced trapping techniques, such as optical tweezers. This would greatly reduce all errors associated with the variance of the atom positions. It would also allow longer trapping times and therefore higher data rates.

Error model

As an (oversimplified) ansatz to estimate the combined effect of the error mechanisms described above, we write the density matrix as a mixture of the ideal density matrix and white noise. This is a common approach to investigate, for instance, the robustness of entanglement witnesses against noise (see, for example, ref. ⁴⁵). The density matrix then reads

in which p_noise is the total error probability, ρ_ideal is the ideal density matrix, ({mathbb{1}}) is the identity matrix and n is the number of qubits. We decompose p_noise into the different error contributions and write

Here p_P denotes the probability of spontaneous scattering during photon emission, p_R the error probability during a Raman rotation, p_F the error probability for the fusion process and N_P, N_R and N_F are the respective number of operations in the protocol. Note that we do not include mechanisms such as decoherence or qubit leakage in the above formula, as we are unable to assign a value to a specific step of the protocol.

In Extended Data Table 2, we compare the fidelity model to the measured lower and upper bounds as defined by equation (20) and equation (26), respectively. For the tree and box graph states, the predicted fidelities ({{mathcal{F}}}_{{rm{model}}}) are found to fall between the measured bounds, as expected. For the hexagon graph, ({{mathcal{F}}}_{{rm{model}}}) falls slightly above the upper bound but is still consistent with it when taking into account the statistical uncertainty (less than one standard deviation). As mentioned earlier, the model does not include the effect of qubit leakage, decoherence and drifts of, for instance, the magnetic field or optical fibres. Hence, it is likely that the predicted fidelities are slightly overestimated.