10-Qubit Entanglement and Parallel Logic Operations with a Superconducting Circuit

Here we report on the production and tomography of genuinely entangled Greenberger-Horne-Zeilinger states with up to ten qubits connecting to a bus resonator in a superconducting circuit, where the resonator-mediated qubit-qubit interactions are used to controllably entangle multiple qubits and to operate on different pairs of qubits in parallel. The resulting 10-qubit density matrix is probed by quantum state tomography, with a fidelity of $0.668\pm 0.025$. Our results demonstrate the largest entanglement created so far in solid-state architectures and pave the way to large-scale quantum computation.

Figure 1

(a) False-color circuit image showing ten superconducting qubits (star shapes) interconnected by a central bus resonator $B$ (gray). Each qubit has its own microwave line (red) for $XY$ control and flux bias line (blue) for $Z$ control, except for $Q_2$ and $Q_6$, which share the microwave lines of neighboring qubits. Each qubit has its own readout resonator, which couples to the circumferential transmission line (orange) for simultaneous readout. (b) Parallel intrapair SE interactions for $Q_3\text{−}{Q}_8$ (top), $Q_5\text{−}{Q}_6$ (middle), and $Q_2\text{−}{Q}_9$ (bottom) at the corresponding detunings as indicated. The anticorrelated, time-modulated occupation probabilities $P_{10}$ (red dots) and $P_{01}$ (blue dots) of each pair indicate that energy is exchanged within the pair [6], undisturbed by what happens in the other two pairs: Six qubits in three pairs are measured simultaneously, and we ignore outcomes of the other qubits for the two-qubit data shown in each panel. All directly measured qubit occupation probabilities are corrected for the elimination of the measurement errors [30]. Lines (green) are numerical simulations. The small high-frequency oscillations in the simulation curve (green) for $Q_3\text{−}{Q}_8$ are due to the relatively small qubit-resonator detuning. These small oscillations can be reduced by using a larger detuning but at the price of a smaller intrapair SE interaction strength.

Figure 2

(a) Pulse sequence with detunings listed in Fig. 1. Tomography is performed to reconstruct the six-qubit density matrix, over which we perform partial trace to obtain the reduced density matrix of each EPR pair. Each $\pi$-rotation (tomographic $\pi /2$-rotation) pulse has a length of 60 (30) ns and a full width half maximum of 30 (15) ns, designed following the derivative reduction by adiabatic gate (DRAG) control theory [49]. (b) Real parts of the reconstructed two-qubit density matrices for the three EPR pairs of $Q_2\text{−}{Q}_9$, $Q_3\text{−}{Q}_8$, and $Q_5\text{−}{Q}_6$, with fidelities (concurrences) of $0.932\pm 0.13$ ($0.869\pm 0.026$), $0.957\pm 0.010$ ($0.915\pm 0.019$), and $0.951\pm 0.010$ ($0.909\pm 0.019$), respectively. For clarity of display, single-qubit $z$-axis rotations are numerically applied to $Q_2$ (93°), $Q_3$ (165°), and $Q_5$ (42°) to cancel the arguments of the major off-diagonal elements.

Figure 3

(a) Pulse sequence for the 10-qubit GHZ state with $\mathrm{\Delta }/2\pi \approx -140\mathrm{MHz}$. (b) Partial elements of the measured 10-qubit density matrix, with a fidelity of $\mathrm{tr}({\rho }_{\text{ideal}}{\rho }_{\exp })=0.668\pm 0.025$ relative to the ideal GHZ state $|{\mathrm{\Psi }}_1⟩=(|{+}_1,{+}_2,\dots ,{+}_N⟩+e^{i\phi }|{-}_1,{-}_2,\dots ,{-}_N⟩)/\sqrt{2}$. For clarity of display, here a single-qubit rotation around the $x^{\prime }$ axis by an angle of $\phi$ is numerically applied to one of the qubits, which cancels the arguments of the dominant off-diagonal elements. Center inset: Cartoon illustration showing ten entangled qubits. Top-left inset: Experimentally measured GHZ fidelity (blue dots) and the data adapted from Ref. [10] (red dots) as functions of the qubit number $N$. Error bars are $1\sigma$. Blue and red dashed lines are guides of different error trends. (c) Parity oscillations observed for the $N$-qubit GHZ states $|{\mathrm{\Psi }}_2⟩$ defined as superpositions of the basis states $|0_1,0_2,\dots ,0_N⟩$ and $|1_1,1_2,\dots ,1_N⟩$ with $N=3–10$. The fringe amplitudes are $0.964\pm 0.016$, $0.956\pm 0.018$, $0.935\pm 0.020$, $0.926\pm 0.026$, $0.796\pm 0.023$, $0.782\pm 0.025$, $0.729\pm 0.028$, and $0.660\pm 0.032$ from top to bottom. For $N=10$, the state preparation and measurement sequence is repeated 81 000 times for a sample size large enough to count the $2^N$ probabilities.