Observation of multi-component atomic Schr\"odinger cat states of up to 20 qubits

TL;DR

This paper reports the deterministic creation of large-scale multi-component atomic Schr"odinger cat states and GHZ states with up to 20 qubits on a superconducting quantum processor, demonstrating advanced control over multipartite entanglement.

Contribution

The work introduces a method to generate and observe multi-component atomic Schr"odinger cat states of up to 20 qubits, the largest entanglement created in solid-state systems to date.

Findings

Successfully generated 18-qubit GHZ state with high fidelity

Created atomic Schr"odinger cat states of up to 20 qubits

Demonstrated controllable multipartite entanglement in a superconducting processor

Abstract

We report on deterministic generation of 18-qubit genuinely entangled Greenberger-Horne-Zeilinger (GHZ) state and multi-component atomic Schr\"{o}dinger cat states of up to 20 qubits on a quantum processor, which features 20 superconducting qubits interconnected by a bus resonator. By engineering a one-axis twisting Hamiltonian enabled by the resonator-mediated interactions, the system of qubits initialized coherently evolves to an over-squeezed, non-Gaussian regime, where atomic Schr\"{o}dinger cat states, i.e., superpositions of atomic coherent states including GHZ state, appear at specific time intervals in excellent agreement with theory. With high controllability, we are able to take snapshots of the dynamics by plotting quasidistribution $Q$-functions of the 20-qubit atomic cat states, and globally characterize the 18-qubit GHZ state which yields a fidelity of $0.525\pm0.005$ confirming genuine eighteen-partite entanglement. Our results demonstrate the largest entanglement controllably created so far in solid state architectures, and the process of generating and detecting multipartite entanglement may promise applications in practical quantum metrology, quantum information processing and quantum computation.

Tables

Table 1. Table S1: Detailed device parameters and qubit performance metrics. ω j 0 superscript subscript 𝜔 𝑗 0 \omega_{j}^{0} is Q j subscript 𝑄 𝑗 Q_{j} ’s maximum resonant frequency at the sweet point, and ω j subscript 𝜔 𝑗 \omega_{j} is Q j subscript 𝑄 𝑗 Q_{j} ’s idle frequency where Q j subscript 𝑄 𝑗 Q_{j} is initialized and single-qubit rotational pulses are applied. T 1 , j subscript 𝑇 1 𝑗 T_{1,j} and T 2 , j ∗ superscript subscript 𝑇 2 𝑗 T_{2,j}^{*} are the typical single-qubit energy relaxation time and Ramsey dephasing time (Gaussian decay)  [ 14 ] , respectively, which are estimated based on the data measured around the interaction frequency where multiqubit GHZ states are generated. Note that T 1 , j subscript 𝑇 1 𝑗 T_{1,j} may fluctuate over time due to the presence of two-level state defects  [ 1 ] and T 2 , j ∗ superscript subscript 𝑇 2 𝑗 T_{2,j}^{*} may not be relevant to the experiment since the qubits coupled with each other at the interacting frequency act as a coherent system which may become insensitive to the local flux noise within individual qubits  [ 28 ] . λ j , j + 1 c superscript subscript 𝜆 𝑗 𝑗 1 𝑐 \lambda_{j,j+1}^{c} describes the crosstalk coupling strength between neighboring qubits, where j 𝑗 j runs cyclically from 1 to 20. g j subscript 𝑔 𝑗 g_{j} is the coupling strength between Q j subscript 𝑄 𝑗 Q_{j} and the bus resonator B 𝐵 B . ω j r superscript subscript 𝜔 𝑗 𝑟 \omega_{j}^{r} is the resonant frequency of the readout resonator for Q j subscript 𝑄 𝑗 Q_{j} . ω j m superscript subscript 𝜔 𝑗 𝑚 \omega_{j}^{m} is the resonant frequency of Q j subscript 𝑄 𝑗 Q_{j} at the beginning of the qubit readout when Q j subscript 𝑄 𝑗 Q_{j} ’s readout resonator is unpopulated. Those values of ω j m superscript subscript 𝜔 𝑗 𝑚 \omega_{j}^{m} for Q 6 subscript 𝑄 6 Q_{6} , Q 10 subscript 𝑄 10 Q_{10} , Q 11 subscript 𝑄 11 Q_{11} , and Q 12 subscript 𝑄 12 Q_{12} in parentheses are used for the N 𝑁 N -qubit GHZ experiment with N 𝑁 N up to 18, during which Q 15 subscript 𝑄 15 Q_{15} and Q 16 subscript 𝑄 16 Q_{16} are biased to ∼ similar-to \sim 4 GHz and can be ignored. F 0 , j subscript 𝐹 0 𝑗 F_{0,j} ( F 1 , j subscript 𝐹 1 𝑗 F_{1,j} ) is the typical probability of detecting Q j subscript 𝑄 𝑗 Q_{j} in | 0 ⟩ ket 0 |0\rangle ( | 1 ⟩ ket 1 |1\rangle ) when it is prepared in | 0 ⟩ ket 0 |0\rangle ( | 1 ⟩ ket 1 |1\rangle ), which is used to correct raw probability data for elimination of the measurement errors  [ 27 ] . Values in parentheses are for the GHZ experiment: As we switched from the 18-qubit GHZ experiment to the 20-qubit atomic Schrödinger cat state experiment, Q 15 subscript 𝑄 15 Q_{15} and Q 16 subscript 𝑄 16 Q_{16} were included so that we reconfigured the ω j m superscript subscript 𝜔 𝑗 𝑚 \omega_{j}^{m} values of 4 qubits, and noticeable drops of a few percent in F 1 subscript 𝐹 1 F_{1} for Q 10 subscript 𝑄 10 Q_{10} , Q 12 subscript 𝑄 12 Q_{12} , and Q 19 subscript 𝑄 19 Q_{19} were captured since we adjusted the working parameters of Josephson parametric amplifiers.

	${\omega }_j^0/2\pi$	${\omega }_j/2\pi$	$T_{1,j}$	$T_{2,j}^{\ast }$	${\lambda }_{j,j+1}^c/2\pi$	$g_j/2\pi$	${\omega }_j^r/2\pi$	${\omega }_j^m/2\pi$	$F_{0,j}$	$F_{1,j}$
	(GHz)	(GHz)	($\mu$s)	($\mu$s)	(MHz)	(MHz)	(GHz)	(GHz)
$Q_1$	5.698	4.320	$\sim$23	2.0	0.75	27.6	6.768	4.510	0.929	0.887
$Q_2$	5.611	4.791	$\sim$27	2.4	0.83	27.4	6.741	4.794	0.969	0.925
$Q_3$	5.793	5.330	$\sim$26	2.0	1.01	29.1	6.707	5.295	0.973	0.920
$Q_4$	5.729	4.865	$\sim$35	1.8	1.02	27.6	6.676	4.491	0.941	0.922
$Q_5$	5.585	4.490	$\sim$30	2.5	-0.39	26.5	6.649	4.435	0.946	0.911
$Q_6$	5.450	4.350	$\sim$29	3.0	1.07	29.2	6.611	4.310 (4.300)	0.927	0.893
$Q_7$	5.480	4.830	$\sim$36	2.7	1.10	27.8	6.589	4.399	0.967	0.885
$Q_8$	5.560	4.965	$\sim$37	2.5	0.83	30.1	6.558	4.905	0.954	0.919
$Q_9$	5.583	4.290	$\sim$20	2.9	0.79	24.1	6.551	4.370	0.933	0.896
$Q_{10}$	5.583	5.290	$\sim$33	2.7	0.65	27.7	6.513	5.375 (5.345)	0.977 (0.967)	0.846 (0.908)
$Q_{11}$	5.682	4.425	$\sim$35	2.8	0.77	27.3	6.524	4.290 (4.340)	0.943	0.889
$Q_{12}$	5.690	5.250	$\sim$33	1.8	0.81	26.9	6.550	5.345 (5.375)	0.981 (0.977)	0.876 (0.903)
$Q_{13}$	5.660	4.899	$\sim$31	2.0	0.96	29.1	6.568	4.819	0.986	0.934
$Q_{14}$	5.723	5.220	$\sim$51	2.4	1.08	27.4	6.598	4.885	0.993	0.951
$Q_{15}$	$\sim$5.7	4.290	$\sim$24	2.1	-0.21	26.3	6.640	4.34	0.981	0.903
$Q_{16}$	5.642	4.260	$\sim$37	2.8	0.77	26.5	6.659	4.01	0.966	0.925
$Q_{17}$	5.843	4.700	$\sim$51	2.3	0.91	27.3	6.685	4.850	0.989	0.940
$Q_{18}$	5.775	4.385	$\sim$37	1.2	0.54	29.0	6.712	4.465	0.967	0.924
$Q_{19}$	5.793	5.170	$\sim$46	2.0	0.67	24.6	6.788	4.930	0.950 (0.988)	0.875 (0.914)
$Q_{20}$	5.847	4.766	$\sim$37	1.7	0.64	27.5	6.758	5.839	0.991	0.813

Equations

$$\begin{gathered} H_1/\hslash ={\omega }_B{a}^{†}a+\sum _{j=1}^{20}\left[{\omega }_j(t)|1_j⟩⟨1_j|+g_j({\sigma }_j^{+}a+{\sigma }_j^{-}{a}^{†})\right] \\ +\sum _{j=1}^{20}{\lambda }_{j,j+1}^{\text{c}}({\sigma }_j^{+}{\sigma }_{j+1}^{-}+{\sigma }_j^{-}{\sigma }_{j+1}^{+}), \end{gathered}$$

$$\begin{gathered} H_2/\hslash =\sum _{\{j,k\}\in N}\frac{g_j{g}_k}{\Delta }\left({\sigma }_j^{+}{\sigma }_k^{-}+{\sigma }_j^{-}{\sigma }_k^{+}\right)+\sum _{j=1}^N\frac{g_j^2}{\Delta }|1_j⟩⟨1_j| \\ +\sum _{j=1}^N{\lambda }_{j,j+1}^{\text{c}}\left({\sigma }_j^{+}{\sigma }_{j+1}^{-}+{\sigma }_j^{-}{\sigma }_{j+1}^{+}\right), \end{gathered}$$

$$\psi (0)=|\theta ,\varphi ⟩={\left[\cos \frac{\theta }{2}|0⟩+\sin \frac{\theta }{2}{e}^{i\varphi }|1⟩\right]}^{\otimes N}.$$

$$\begin{gathered} \psi \left(t=\pi /2|\lambda |\right)=e^{-iHt}|\theta ,\varphi ⟩=\frac{1}{\sqrt{2}}{e}^{-i(N-0.5)\pi /2} \\ \times \left[|\theta ,\varphi -\frac{N-1}{2}\pi ⟩+e^{-i\pi /2}|\theta ,\varphi -\frac{N-3}{2}\pi ⟩\right]. \end{gathered}$$

Full text

The capability of controllably entangling multiple particles is central to fundamental test of quantum theory [1], and represents a key prerequisite for quantum information processing. There exist various kinds of multipartite entangled states, among which the Greenberger-Horne-Zeilinger (GHZ) states, i.e., the 2-component atomic Schrödinger cat states, are particularly appealing and useful [2]. These states play a key role in quantum-based technologies, including open-destination quantum teleportation [3], concatenated error correcting codes [4], quantum simulation [5], and high-precision spectroscopy measurement [6]. In principle, the number of particles that can be deterministically entangled in a quantum processor is a benchmark of its capability in processing quantum information. However, it is difficult to scale up this number since the conventional step-by-step gate methods require long control sequences which increase exposure to perturbing noise. A shortcut is to realize the free evolution under a nonlinear Hamiltonian with, e.g., one-axis twisting, and the system of qubits initialized in an atomic coherent state is predicted to evolve to squeezed spin states [7], and then to the multi-component atomic Schrödinger cat states [8], i.e., superpositions of atomic coherent states including GHZ state [9].

Engineering fully controllable and highly coherent multipartite quantum computing platforms remains an outstanding challenge. Several physical platforms are being explored [11, 10, 12, 13, 14, 15], and a series of experiments for generating multipartite entanglement were reported [16, 17, 18, 14, 15, 19, 20, 21, 22, 23]. Some of these experiments involve local detections of only the subsystems [15, 19]. Multipartite entanglement, in particular the GHZ state which possesses global entanglement, would be better characterized by synchronized detections of all system parties and was achieved with 14 trapped ions [20], 12 photons [21], 18 photonic qubits exploiting 6 photons [22], and 12 superconducting qubits [23]. In particular, previously we reported the production and full tomography of the 10-qubit GHZ state and the implementation of high-fidelity two-qubit gate with an all-to-all connected superconducting quantum processor [14, 24], where each qubit can be individually controlled and qubit-qubit interactions can be turned on and off as desired.

In this letter we introduce our latest upgrade, a more powerful 20-qubit superconducting quantum processor featuring all-to-all connectivity with programmable qubit-qubit couplings mediated by a bus resonator. With all qubits designed to be uniformly coupled to the bus resonator, we engineer a one-axis twisting Hamiltonian by identically detuning the qubits from the bus resonator. Free evolution under the engineered Hamiltonian steers the system to squeezed spin states, and then to over-squeezed regime with suppositions of atomic coherent states at specific time intervals, which are experimentally captured. The final GHZ states are characterized by synchronized local manipulations and detections of all qubits, and we measure a fidelity figure of $0.525\pm 0.005$ for 18 qubits, which confirms the genuine eighteen-partite entanglement [25].

The new version of the superconducting quantum processor and critical peripheral electronics are illustrated in Fig. 1(a), which consists of 20 frequency-tunable transmon qubits, labeled as $Q_{j}$ for $j$ = 1 to 20, surrounding a central coplanar waveguide bus resonator ($B$), whose resonant frequency is fixed at $\omega_{B}/2\pi\approx$ 5.51 GHz. Qubit-resonator ($Q_{j}$-$B$) coupling strengths $g_{j}$ are designed to be uniform, and measured $g_{j}/2\pi$ values range from 24.1 to 30.1 MHz. Qubits are detected through their respective readout resonators, whose signal spectra are shown in Fig. 1(b). We use impedance matched Josephson parametric amplifiers (JPAs) and an optimized arrangement of the qubit frequencies, $\omega_{j}^{m}$, during the readout to enhance the signal-to-noise ratio.

All qubits are individually tunable with high flexibility, and we show an example in Fig. 1(c) by measuring $Q_{20}$’s swap spectroscopy while we equally space the other 19 qubits in frequency around the resonator $B$. Typical qubit energy relaxation times, $T_{1}$, are in the range of 20 to 50 $\mu$s. With a proper arrangement of the qubit idle frequencies, $\omega_{j}$, where qubit initializations and single-qubit rotations are applied, fidelity values of the simultaneous single-qubit $\pi/2$ rotational gates used in the GHZ experiment are all above 0.99 as estimated by quantum state tomography and simultaneous randomized benchmarking. See Supplemental Material for more details on the device and its operations [26].

With each of the 20 qubits being addressable, the system Hamiltonian is

[TABLE]

where $\omega_{j}(t)$ ($\gg g_{j}$) is tunable within a time scale of a few nanoseconds, $\sigma_{j}^{+}$ ($\sigma_{j}^{-}$) is the raising (lowering) operator of $Q_{j}$, $a^{\dagger}$ ($a$) is the creation (annihilation) operator of $B$, and $\lambda^{\textrm{c}}_{j,j+1}$ describes the crosstalk couplings between neighboring qubits (Subscripts in $\lambda^{\textrm{c}}_{j,j+1}$ run cyclically from 1 to 20). Although more qubits are integrated in this processor, the measured $\lambda^{\textrm{c}}_{j,j+1}/2\pi$ values are seen to be reduced from $\sim$2 MHz in the previous 10-qubit version [14, 28] to around 1 MHz or less since we separate the qubits physically as much as possible (see Fig. 1(a)). Note that there may exist those qubit-qubit crosstalk couplings beyond neighboring pairs, which should be relatively small and are not included in Eq. (1).

As demonstrated previously [14], the unique feature of this architecture is that, although qubits are physically separated by the bus resonator $B$, the qubit-qubit coupling mediated by $B$ can be programmed with fast Z controls to match or detune their frequencies [14, 24]. More remarkably, in our processor, we can selectively entangle $N$ of the 20 qubits by detuning the selected qubits from the resonator by the same amount $\Delta$ ($\gg g_{j}$), with the other qubits being far off-resonant. When resonator $B$ is initially in vacuum, the effective Hamiltonian for these $N$ qubits, relabeled by $Q_{j}$ with $j$ going from 1 to $N$, in the frame rotating at the detuned qubit frequency is [9, 8]

[TABLE]

where $\{j,k\}$ takes all possible pairs within the $N$ qubits and subscripts in $\lambda^{\textrm{c}}_{j,j+1}$ run cyclically from 1 to $N$.

The scenario of a system of $N$ identical two-level atoms interacting collectively and dispersively with a single mode electromagnetic field in a cavity has been theoretically investigated [8, 9]. In our experiment we position the qubits 330 MHz below $\omega_{B}/2\pi$ for all the effective qubit-qubit couplings (190 terms) in the first summation of Eq. (2), $\left|{g_{j}g_{k}}/{2\pi\Delta}\right|$, to be $\sim 2$ MHz while the few ($<20$ terms) neighboring couplings $\lambda^{\textrm{c}}_{j,j+1}/2\pi$ are from 0.5 to 1 MHz. Therefore we can ignore $\lambda^{\textrm{c}}_{j,j+1}$ and those relatively small qubit-qubit crosstalk couplings beyond neighboring pairs (not included in Eq. (1)) for now and assume that couplings within all qubit pairs are approximately equal, so that the theory predictions [8, 9] can be adapted to our experiment. We emphasize that the imperfection in uniformity has been taken into account by numerical simulations using device parameters based on the Hamiltonian in Eq. (1), and we find decent agreement between our experimental results and the simplified theoretical treatment in Refs. [8, 9].

With uniform couplings noted as $\lambda=\overline{g_{j}g_{k}}/{\Delta}$, we now apply the spin representation of qubit states and define the collective spin operators $\mathcal{S}^{+}=\sum_{j}{\sigma_{j}^{+}}$, $\mathcal{S}^{-}=\sum_{j}{\sigma_{j}^{-}}$, and $\mathcal{S}_{z}=\sum_{j}{\sigma_{z,j}}$. The term $\sum{\lambda(\sigma_{j}^{+}\sigma_{k}^{-}+\sigma_{j}^{-}\sigma_{k}^{+})}$ in Eq. (2) is then transformed to $\lambda\mathcal{S}^{+}\mathcal{S}^{-}\rightarrow-\lambda\mathcal{S}_{z}^{2}$ ignoring trivial linear and constant terms, which is the one-axis twisting Hamiltonian. By initializing the $N$ qubits identically so that each individual qubit points to the same direction represented by the angles ($\theta$, $\phi$) in its Bloch sphere, we write down the wavefunction of the atomic (spin) coherent state as

[TABLE]

Evolution of the wavefunction under the one-axis twisting Hamiltonian, $H=-\lambda\mathcal{S}_{z}^{2}$, was analytically obtained in Ref. [8], which shows that at particular time $t=\pi/m\lambda$, where $m$ is an integer no less than 2, $\psi(0)$ evolves to a superposition of multiple atomic coherent states, i.e., it becomes an atomic Schrödinger cat state. In particular, at $m=2$, it evolves to a superposition of two atomic coherent states, i.e., the $N$-qubit GHZ state,

[TABLE]

Figure 2(a) shows the pulse sequence for generating and characterizing the $N$-qubit GHZ state. We start with initializing each of the $N$ qubits in $\left(\ket{0}-i\ket{1}\right)/\sqrt{2}$, which collectively corresponds to an atomic coherent state $|\pi/2,-\pi/2\rangle$ in the ($\theta$, $\phi$) notation, by applying an $X_{\pi/2}$ rotational pulse at the qubit’s idle frequency (sinusoids in zone I), following which we bias the $N$ qubits to $\Delta/2\pi\approx-330$ MHz for an optimized duration close to $\pi/2|\lambda|$ (zone II). The phase of each qubit’s XY drive, which defines the rotational axis in the equator plane, is calibrated according to the rotating frame with respect to $\Delta$, ensuring that all $N$ qubits are in the same initial state just before their collective interactions are switched on [14, 5]. Right after the interactions we bias these qubits back to their respective idle frequencies, $\omega_{j}$, for further operations if necessary, and then to their respective measurement frequencies, $\omega_{j}^{m}$, for readout. We note that during the frequency tuning process qubits may gain different dynamical phases, i.e., the $x$-$y$ axes rotate differently in the equator planes for different qubits, which can be determined by a separate phase tracking measurement followed by an optimization procedure (see Supplemental Material [26]).

The resulting GHZ state is a superposition of $\left|\pi/2,-N\pi/2\right\rangle$ and $\left|\pi/2,-(N-2)\pi/2\right\rangle$ in the collective spin representation, which can be transformed to a superposition of the $N$ qubits all in $|0\rangle$ and those all in $|1\rangle$ by applying to each qubit a $\pi/2$ rotation around its $x$ ($N$ odd) or $y$ ($N$ even) axis. After such a transformation (sinusoids in zone III of Fig. 2(a)), the wavefunction is written as $\left(|00...0\rangle+e^{i\varphi}|11...1\rangle\right)/\sqrt{2}$, where $\varphi=\pi/2$ for uniform couplings. The diagonal elements of the GHZ density matrix $\rho_{00...0}$ and $\rho_{11...1}$ can be directly probed: For each state generation and characterization pulse sequence we simultaneously measure all qubits which returns an $N$-bit binary string, e.g, 01…0, showing the collapsed multiqubit state; we repeat the same pulse sequence multiple times and count the probabilities of finding all $N$ bits in 0 for $\rho_{00...0}$ and all those in 1 for $\rho_{11...1}$.

The off-diagonal elements $\rho_{00...0,\,11...1}$ and $\rho_{11...1,\,00...0}$ can be obtained by measuring the parity oscillations, defined as the expectation value of the operator $\mathcal{P}(\gamma)=\otimes_{j=1}^{N}(\cos{\gamma}Y_{j}+\sin{\gamma}X_{j})$, which is given by $\langle\mathcal{P}(\gamma)\rangle=2\left|\rho_{00...0,\,11...1}\right|\cos(N\gamma+\varphi)$ for the abovementioned GHZ wavefunction [20]. Experimentally we apply to each qubit a rotation (sinusoids in zone IV of Fig. 2(a)) which bring the axis defined by the operator $\mathcal{P}(\gamma)$, i.e., the direction represented by the angles ($\pi/2$, $\pi/2-\gamma$) in each qubit’s Bloch sphere, to the $z$ axis, followed by simultaneous qubit readout. Repeating each state generation and measurement pulse sequence multiple times yields $2^{N}$ probabilities ($P_{00...0}$, $P_{00...1}$, …, $P_{11...1}$), and the parity is calculated as $\langle\mathcal{P}\rangle=P_{\textrm{even}}-P_{\textrm{odd}}$ with $P_{\textrm{even}}$ ($P_{\textrm{odd}}$) corresponding to the summation of all those probabilities with even (odd) number of qubits in $|1\rangle$. The clear oscillation patterns of $\langle\mathcal{P}(\gamma)\rangle$, whose amplitude gives $\left|\rho_{00...0,\,11...1}\right|$, confirm the existence of coherence between the two states $|00...0\rangle$ and $|11...1\rangle$ (Fig. 2(b)). Using values of $\rho_{00...0}$, $\rho_{11...1}$, and $\left|\rho_{00...0,\,11...1}\right|$ obtained above, $N$-qubit GHZ state fidelities are calculated to be $0.817\pm 0.009$ ($N=10$), $0.775\pm 0.011$ ($N=12$), $0.655\pm 0.009$ ($N=14$), $0.579\pm 0.007$ ($N=16$), $0.549\pm 0.006$ ($N=17$), and $0.525\pm 0.005$ ($N=18$), all confirming genuine multipartite entanglement [25].

Furthermore, detailed dynamics connecting the atomic coherent state in Eq. (3) to the GHZ state in Eq. (4) under the one-axis twisting Hamiltonian was analytically given in Ref. [8], where squeezed spin states and more atomic Schrödinger cat states other than the final GHZ state sequentially appear. We are able to take snapshots of this dynamic process with up to 20 qubits by measuring the quasidistribution $Q$-function $Q(\theta,\phi)\propto\langle\theta,\phi|\rho(t)|\theta,\phi\rangle$, where $\rho(t)$ is the evolving multiqubit density matrix. For the 20 qubit case, we bias all qubits to $\Delta/2\pi\approx-470$ MHz since a newly added qubit, $Q_{15}$, is interfered by a two-level state defect at the previous entangling frequency. To obtain $Q(\theta,\phi)$, we rotate the axis defined by the angles ($\theta$, $\phi$) to the $z$ axis for each qubit and do so simultaneously for all $N$ qubits before joint readout: For $\theta<\pi/2$, we rotate by angle $-\theta$ around the $\phi-\pi/2$ axis in the equator plane and record $P_{00...0}$ as $Q_{\textrm{exp}}$; for $\theta>\pi/2$, we rotate by angle $\pi-\theta$ around the $\phi+\pi/2$ axis in the equator plane to reduce the amplitude of the rotational pulse and record $P_{11...1}$ as $Q_{\textrm{exp}}$. Obtained values of $Q_{\textrm{exp}}$ are plotted as functions of $\theta$ and $\phi$ in the spherical polar plots as shown in Fig. 3, together with the numerical simulations ignoring decoherence. We observe the squeezed spin regime at the beginning ($\sim 15$ ns) and the atomic Schrödinger cat states which are superpositions of $m=5$, 4, 3, and 2 atomic coherent states at $t_{m}\approx 80$, 95, 130, and 195 ns, respectively. It is seen that $t_{m}\propto 1/m$ as predicted in Ref. [8]. For an $m$-component atomic Schrödinger cat state of $N$ qubits, the overlap between adjacent two components is $\cos^{N}(\pi/m)$. Therefore to observe superpositions with more components one needs to increase $N$ to reduce the overlap. We note that superpositions of up to 4 coherent states have been previously observed in cold atoms and superconducting cavities [29, 30, 31]. Here for the first time we observe the 5-component atomic Schrödinger cat state with $N=20$ qubits.

In summary, our experiment demonstrates an upgraded and much more powerful version of the multiqubit-resonator-bus architecture for scalable quantum information processing, with 20 individually addressable qubits and programmable qubit-qubit couplings. Based on this device, we efficiently and deterministically generate the 18-qubit genuinely entangled GHZ state and multi-component atomic Schrödinger cat states of up to 20 qubits by engineering a one-axis twisting Hamiltonian. The high controllability and efficiency of our superconducting quantum processor demonstrate the great potential of an all-to-all connected circuit architecture for scalable quantum information processing.

Acknowledgments. This work was supported by the National Basic Research Program of China (Grants No. 2017YFA0304300 and No. 2016YFA0300600), the National Natural Science Foundations of China (Grants No. 11725419 and No. 11434008), and Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB28000000). Devices were made at the Nanofabrication Facilities at Institute of Physics in Beijing and National Center for Nanoscience and Technology in Beijing.