Generating scalable graph states in an atom-nanophotonic interface

In an atom-nanophotonic interface with a single-sided cavity, Bell states can be prepared via state carving technique [24–26] with contrasted single-photon reflectivity spectra and can be transported to distant places [45], promising a scalable quantum network. The state carving protocol starts from the initialized states $|++\rangle$ with N = 2 atoms via single-qubit rotations between the atomic coupled state $|1\rangle = |g\rangle$ and the uncoupled state $|0\rangle$ as qubit spaces, where $|+\rangle\equiv(|0\rangle+|1\rangle)/\sqrt{2}$. This leads to a full expansion of four atomic states $(|00\rangle+|01\rangle+|10\rangle+|11\rangle)/2$ with the number of coupled atoms N_a = 0, 1, 1, and 2, respectively. Here we denote N_a as the number of atoms in the coupled state $|1\rangle$ out of a total number N of atoms involved. Through an atom-cavity coupling at the atomic transition $|g\rangle\rightarrow|e\rangle$ and a relatively high single-photon reflectivity R for at least one of the atoms in the coupled states compared to a low R for $|00\rangle$, a weak-field probe projects $|++\rangle$ into $(|01\rangle+|10\rangle+|11\rangle)/\sqrt{3}$ upon a photon detection. This effectively carves out the unwanted component $|00\rangle$ and heralds the bipartite entanglement. The other unwanted component $|11\rangle$ can be removed as well by another state carving, after single-qubit rotations $|0\rangle\leftrightarrow|1\rangle$ on both qubits, upon again detecting a second probe photon. This two-stage state carving protocol effectively generates an entangled state $(|01\rangle+|10\rangle)/\sqrt{2}$ with a probability up to $1/2$ [24] and an optimal state fidelity depending on the maximal and minimal values of R [45].

Here we consider an atom-nanophotonic interface in a single-sided cavity as shown in figure 1. The Hamiltonians for this system can be written as ($\hbar$ = 1 and see supplementary note 1)

$$\begin{align} H_{s} & = \omega_{c}a^{\dagger}a+\omega_a\sum_{\mu = 1}^{N_a}\sigma_\mu^{\dagger}\sigma_\mu+\sum_{\mu = 1}^{N_a} g_\mu\cos\left(k_sx_\mu\right)\left(a^{\dagger}\sigma_\mu+\sigma_\mu^{\dagger}a\right), \end{align} \tag{ 1 }$$

where the atoms are placed at the antinodes of the cavity fields with a profile $\cos(k_sx_\mu)$ for significant atom-cavity coupling strengths. The system dynamics of the density matrix ρ can be solved from $\dot{\rho} = -i[H_s,\rho]+\gamma \sum_{\mu = 1}^{N_a} D[\sigma_\mu]\rho+\kappa D[a]\rho$. The cavity resonance frequency is ω_c with a photon operator a, the atomic transition frequency is ω_a with dipole operators $\sigma_\mu\equiv|g\rangle_\mu\langle e|$ for the ground state $|g\rangle$ and the excited state $|e\rangle$, and an atom-photon coupling constant is g. A number of N_a atoms are coupled with the nanophotonic cavity fields via the evanescent waves above the surface of the waveguide [30, 34], where k_s denotes the wave vector of the guided mode and $D[c]\rho\equiv c \rho c^\dagger-\{c^\dagger c,\rho\}/2$ with $c\in\{a, \sigma_\mu\}$. The total decay rate of the atom is $\gamma\equiv 2|dq(\omega)/d\omega|_{\omega = \omega_a}g_{k_s}^2L$, with $|dq(\omega)/d\omega|$ the inverse of group velocity, $q(\omega)$ the resonant wave vector, the coupling strength $g_{k_s}$, and the quantization length L. A measure of single-atom cooperativity $C\equiv 4g^2/(\kappa\gamma)$ further identifies a strong-coupling regime $C\gg 1$ [18] under a total cavity decay rate $\kappa = \kappa_\textrm{wg}+\kappa_\textrm{sc}$ involving the decay rate to the waveguide (wg) and the nonguided rate for scattered (sc) light. This atom-nanophotonic interface can be implemented either with trapped atoms in optical lattices [28] or in optical tweezer arrays [18, 45] as shown schematically in figure 1.

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The single-photon reflectivity $R = |r|^2 = |a_\textrm{out}/a_\textrm{in}|^2$ can be obtained by solving the steady-state solutions of $\langle A\rangle = \textrm{Tr}[\rho(t\rightarrow\infty) A]$ where $A\in\{a,\sigma_1,\sigma_2,{\ldots},\sigma_{N_a}\}$ based on the input–output formalism [47] under a weak excitation limit ($|g\rangle_\mu\langle g|\approx 1$) with $a_\textrm{out}+a_\textrm{in} = \sqrt{\kappa_\textrm{wg}}a$ (see appendix and supplementary note 1). In figure 2, we present the on-resonance single-photon reflectivity R for two coupled atoms $N_a = 2$ and the reflectivity spectrum for different N_a . We find that when a critical coupling regime is reached [18], that is $\kappa_\textrm{wg} = \kappa_\textrm{sc}$ (supplementary note 2), a vanishing R emerges (figures 2(a) and (b)). This suggests the most contrasted single-photon reflectivity which can be utilized to conduct the state carving and project the target states with high fidelity. Meanwhile, this critical coupling regime can be manipulated and controlled by the number of air holes in the region of mirror in the photonic crystal cavity [18, 48, 49]. In particular, a pattern of 500 devices have been created, where a range of loss rates from 2 to 30 GHz can be characterized depending on the number of air holes in the mirror [18]. This offers a versatile platform with wide ranges of nanophotonic cavity properties and makes possible the critical coupling regime we require in this work.

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We note that the on-resonance R in figure 2(a) are robust to the position variations from the anti-nodes of the cavity field profile in the strong coupling regime. On the other hand in the moderate coupling regime, a sensitivity to position fluctuations around the anti-nodes $k_sx_\mu = \pm\pi$ arises [18], which leads to a degradation in the generation probability of graph states but not the fidelity. This fragility can be mitigated by lower the temperature of the atoms in optical tweezers via Raman sideband cooling [50] or gray-molasses loading to optical tweezers [51]. While an adiabatic transport of atoms can manipulate the coupling or uncoupling between the atoms and the cavity photon, which may suffer from atom losses or decoherences [45], an extra qubit state $|2\rangle$ uncoupled to the cavity can be utilized and transferred from $|1\rangle$ to decouple from the cavity. This only requires a single-qubit gate between $|1\rangle$ and $|2\rangle$, and should be more resilient to adiabatic transfer errors.

With the contrasted single-photon reflectivity spectrum in the atom-nanophotonic cavity interface, here we propose to generate a special class of multipartite entangled states including linear and two-dimensional cluster states (figure 3). These entangled states establish the foundations for measurement-based quantum computation [3–5, 52, 53]. A graph state is a multipartite entangled state associated with a graph $G(V,E)$, where V and E stand for the sets of vertices and edges, respectively. The following definition shows the general expression of graph states $|G\rangle$ [2] as

$$\begin{align} |G\rangle\equiv \Pi_{\left(a, b\right)\in E}CZ\left(a,b\right)|+\rangle^{\otimes |V|}, \end{align} \tag{ 2 }$$

where $CZ(a,b)$ represents the controlled-Z (CZ) gate between qubit spaces spanned by two nodes or vertices a and b linked by edges. Meanwhile, a cluster state denotes a special case of graph states, whose associated graph is a connected subset on cubic lattices [2]. Alternatively, the graph states can be defined equivalently as the common eigenstates of a set of stabilizers $K_a = \sigma_x^a\Pi_{b\in N_G(a)}\sigma_z^b$ with all eigenvalues of one for all $a\in V$. Here σ_x and σ_z are Pauli matrices and N_G represents the set of vertices whose elements are in the neighborhood of the vertex a. In terms of the stabilizer formalism, the graph states provide useful and crucial resources for quantum error correction [54].

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Based on the definition of cluster states in the conventional graph representations, we demonstrate some examples of linear cluster states. For two-node and three-node linear cluster states, they are $(|+0\rangle+|-1\rangle)/\sqrt{2}$ and $(|+0+\rangle+|-1-\rangle)/\sqrt{2}$, respectively, as $|\pm\rangle\equiv(|0\rangle\pm|1\rangle)/\sqrt{2}$, and CZ gate imprints an extra π phase conditional on the two-qubit states $|11\rangle$, transforming $|1\pm\rangle$ to $|1\mp\rangle$. For four-node linear cluster state, it can be expressed as $|\psi\rangle_{LCS}$ = $(|+0+0\rangle+|+0-1\rangle+|-1-0\rangle+|-1+1\rangle)/2$, where a notation of $|(\textrm{sign})(\textrm{digit})(\textrm{sign})(\textrm{digit}){\ldots}\rangle$ is adopted. Before applying the state carving (SC) protocol for preparing scalable linear graph states $|\textrm{LGS}\rangle$, it would be convenient to rotate them to what we dub the 'precursor' state $(|1010\rangle+|1001\rangle+|0100\rangle+|0111\rangle)/2$ by single-qubit rotations R on the qubit number 1 and 3, respectively (supplementary note 3).

The conventional linear cluster state generation requires a CZ gate on two qubits linking one of the nodes denoted by a in the initial graph state $|\textrm{G}\rangle$ with an extra qubit $|+\rangle_n$. The resultant new graph state becomes

$$\begin{align} \textrm{CZ}\left(a,n\right)|\textrm{G}\rangle|+\rangle_n = \frac{1}{\sqrt{2}}\left(|\phi_0\rangle|0\rangle_a|+\rangle_n+|\phi_1\rangle|1\rangle_a|-\rangle_n\right), \end{align} \tag{ 3 }$$

where $\textrm{CZ}(a,n)$ represents the CZ gate operation and the states $|\phi_{0(1)}\rangle$ are the graph state components associated with the qubit node a. The resultant graph state is the target state for our proposed protocols using SC in an atom-nanophotonic interface. With a vanishing R for the states $|00\rangle$, we obtain the projected and normalized states after SC(a, n) between the qubits a and n upon single-photon reflection,

$$\begin{align} \textrm{SC}\left(a,n\right)|\textrm{G}\rangle|+\rangle_n& = \textrm{SC}\left(a,n\right)\left(|\phi_0\rangle|0\rangle_a|0\rangle_n+|\phi_0\rangle|0\rangle_a|1\rangle_n+|\phi_1\rangle|1\rangle_a|0\rangle_n+|\phi_1\rangle|1\rangle_a|1\rangle_n\right),\nonumber\\ & \rightarrow \frac{1}{\sqrt{3}}\left(|\phi_0\rangle|0\rangle_a|1\rangle_n+|\phi_1\rangle|1\rangle_a|0\rangle_n+|\phi_1\rangle|1\rangle_a|1\rangle_n\right), \end{align} \tag{ 4 }$$

where $|1\rangle_a|1\rangle_n$ can be rotated to $|0\rangle_a|0\rangle_n$ by single-qubit operations and carved out again using SC, transforming equation (4) to the target state of equation (3) up to single-qubit rotations as shown in figure 3(a) (supplementary note 4). This lays the foundation of the protocol for any linear cluster states by weaving the cluster state with extra qubits successively [$2(M-1)$ times of SC for M nodes] and for certain two-dimensional graph states like a cross with five nodes or horseshoe cluster states [4].

To weave general two-dimensional cluster states, we need multiqubit SC on qubits more than $N_a = 2$ (figures 3(b) and (c)). We take $N_a = 3$ in figure 3(b) as an example, and the target state to link the extra qubit $|+\rangle_n$ with two edges and two nodes (a and b) in the graph state becomes

$$\begin{align} \textrm{CZ}\left(a,n\right)\textrm{CZ}\left(b,n\right)|\textrm{G}\rangle|+\rangle_n = \frac{1}{2}\left(|\phi_{00}\rangle|00\rangle_a|+\rangle_n + |\phi_{01}\rangle|01\rangle_a|-\rangle_n + |\phi_{10}\rangle|10\rangle_a|-\rangle_n + |\phi_{11}\rangle|11\rangle_a|+\rangle_n\right). \end{align} \tag{ 5 }$$

Via four times of SC on the nodes a, b, and n to remove the state component $\rangle|00\rangle_a|0\rangle_n$, followed by X gates on (a, b) and (b, n) respectively and successively as in figure 3(b), we obtain (supplementary note 4)

$$\begin{align} \frac{1}{2}\left(|\phi_{00}\rangle|00\rangle_a|1\rangle_n + |\phi_{01}\rangle|01\rangle_a|0\rangle_n + |\phi_{10}\rangle|10\rangle_a|0\rangle_n + |\phi_{11}\rangle|11\rangle_a|1\rangle_n\right), \end{align} \tag{ 6 }$$

which is exactly the target precursor state and then can be transformed to the target graph state by applying a single-qubit rotation on the nth qubit, fulfilling the protocol to generate graph states more than one dimension (figures 3(b) and (c)). In general, an arbitrary M links with $|+\rangle_n$ can be achieved by applying the similar designs in the protocols of figure 3. Two arbitrary graph states, $|\textrm{G}_1\rangle$ and $|\textrm{G}_2\rangle$, can also be combined by SC on an additional qubit $|+\rangle_n$ and two nodes in respective graph states. This can be extended to combine arbitrary number of M graph states into a larger graph state with again SC on an additional $|+\rangle_n$ and M nodes with 2^M numbers of SC (supplementary note 4).

One of the basic elements in two-dimensional graph states is the square graph state. This can be weaved from two one-link qubits and one two-link qubit (figure 4), which results in $(|0+0+\rangle+|0-1-\rangle+|1-0-\rangle+|1+1+\rangle)/2$. A generic two-dimensional graph state at arbitrary sizes and shapes can then be realized with assistance of multiqubit SC as demonstrated in figure 3. On the other hand, linear graph states can be created only by using 2-atom SC (figure 3(a)), but more exquisite protocols can as well be developed involving multiqubit SC that can improve the optimal probability to generate them, which we discuss more in the next section and in supplementary note 4.

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In the following, we investigate the probability and the fidelity of graph state preparation, which can provide insights into the capabilities and limits of the proposed protocols. We also propose an improving protocol by applying a sequential single-photon probe to create optimal probability, high-fidelity, and scalable graph states. A trade-off for our protocol in figure 3(a) is the reduction of the generation probability with the size of the graph state, as more unwanted states need to be carved away. In this regard, we utilize either the multiqubit SC between one node and $|+\rangle^{\otimes 2}$ or multiqubit SC among $|+\rangle^{\otimes N}$ specifically with the digit-sign representation (alternative to sign-digit representation) in the target linear graph states, which optimizes the probability of state preparations with less state components to be carved out (supplementary note 4). We compare the probability and fidelity for linear graph state generation by 2-atom (one node and $|+\rangle$) and 3-atom state carving (one node and $|+\rangle^{\otimes 2}$) protocols in figure 5. The maximum probability P of generating linear graph states in each protocol can be achieved for the ideal case of an infinite cooperativity (i.e. $C = \infty$ and $R_{N_a\unicode{x2A7E} 1} = 1$), which yields $P = 2^{-(N-1)}$ and $2^{-\lfloor N/2\rfloor}$ with a floor function of $N/2$, respectively, showing the advantage of multiqubit SC. In practice, a finite C would produce a non-unity reflectivity R. For a number of $N_\mathrm{SC}$ state carvings, a factor of $R^{N_\mathrm{SC}}$ will be multiplied to the probability which quickly decreases as N increases (red squares in figure 5 for C = 20 as an example).

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To achieve the maximum probability, we would need to modify the state carving procedure by sending multiple single photons sequentially for each state carving instead of using just one photon each time. This procedure will dramatically decrease the deteriorating effect of non-unity R. For example, the total probability of getting a photon in reflection for Bell state-carving will rise from $P_{01}+P_{10} = R_1/2$ to $[1-(1-R_1)^{N_p}]/2$ for N_p sequential photons (supplementary note 5), where $P_{01(10)}$ denotes the weighted number of components in the states $|01(10)\rangle$ within $|+\rangle^{\otimes 2}$ upon single-photon detection. If $R_1 = 0.8$, with just $N_p = 2$ we have $1-(1-R_1)^{N_p} = 0.96$ which is near unity, and this dramatically improves the probability to nearly reach the optimal probabilities at an infinite C. For three-atom state carving in figure 5(b), the sequential probability at a finite C is estimated by choosing $R_3\rightarrow R_1$, showing significant improvement even with an underestimated R₃.

The fidelity of the generated graph states is a deciding factor for measurement-based quantum applications. A high-fidelity graph state indicates genuine multipartite entanglement, which is essential as a resource for one-way quantum computation. The fidelity for generating arbitrarily large graph states becomes perfect when $\kappa_\mathrm{wg} = \kappa_\mathrm{sc} = \kappa/2$ as the critical coupling regime with r₀ = 0 (blue circles in figure 5). We note that r₀ is the probability of reflection when no atom is coupled to the cavity. All state carvings intend to remove these uncoupled states (i.e. $|00\rangle$, $|000\rangle$, or $|0000\rangle$ states etc). Hence, if r₀ = 0, no unwanted state component is projected out when a photon is reflected, resulting in a unit fidelity of the desired state. If $r_0 \neq 0$, some unwanted state contribution emerges, and the fidelity deteriorates as seen in the case of $\kappa_\mathrm{wg}/\kappa = 0.4$ (green circles in figure 5). However, these fidelities can also be improved arbitrarily close to 1 when multiple sequential photons are used. Furthermore, the fidelity is resilient to uneven atom-photon couplings (i.e. $g_i \neq g_j$ or equivalently $C_i \neq C_j$ for ith and jth atoms). Therefore, our proposed protocol is robust against multiple forms of imperfections of $g_i \neq g_j$, $r_0 \neq 0$ away from the critical coupling regime, and $|r_1| \lt 1$ for a finite C (supplementary note 5).

As a final remark, we note that the non-unit detector efficiency and dark counts from environment would affect the results presented here. The non-unit detector efficiency would lessen the probability of graph state preparations, while the unwanted dark counts would undermine the fidelity of the generated graph states. Nevertheless, the total detection efficiency is mainly limited by the quantum efficiency of single-photon detector and the fiber taper coupling efficiency, which can be further improved by using superconducting nanowire detectors and a higher fiber coupling efficiency [45], respectively. For the effect of dark counts, a lower ambient temperature can be applied in the system to suppress their influences. We note that the non-unit detection efficiency corresponds to an effectively lower reflectivity R, which can nonetheless be improved by our protocol using multiple single photon probes discussed earlier. Therefore, both of these imperfect operational factors can be made unsubstantial with technical efforts, and our results can be kept intact promisingly.