All-microwave nonadiabatic multiqubit geometric phase gate for superconducting qubits

Geometric phase gates are promising tools toward robust quantum computing owing to their robustness against certain control errors and decoherence. Here, we propose a multiqubit architecture with a nonadiabatic geometric phase gate scheme that is feasible in widely used superconducting qubit designs such as transmon and fluxonium. Through segmented microwave drive on multiple qubits coupled off-resonantly to a common resonator, the geometric phase gate is obtained from the qubit-state-dependent displacement of the resonator without fine-tuning the qubit frequencies. Fidelity above 99.99% is achieved in simulation under the available experimental parameters. Our scheme uses all-microwave control and only exploits the lowest qubit levels with long coherence time; thus it is desirable for experiments. Together with the single-qubit holonomic gates demonstrated in earlier experiments, our scheme can realize universal all-geometric quantum computing, and it also finds applications in quantum simulation with many-body interactions.

Figure 1

(a) The system consists of $n$ superconducting qubits ($Q_1,Q_2,...,Q_n$) coupled to a common microwave resonator (bus). (b) A schematic of the two-qubit geometric phase gate. The two target qubits are selectively driven by a microwave pulse ${\mathrm{\Omega }}_i\left(t\right)cos{\omega }_{d}t$ ($i=1,2$) via the XY control line.

Figure 2

(a) Schematic pulse sequence for the ideal two-qubit geometric phase gate. (b) The qubit-state-dependent phase-space trajectories of the resonator mode. (c) The ideal time evolution of the system based on the Hamiltonian in Eq. (2) without the last term starting from the initial state $\left(\right|g\rangle +|e\rangle )\left(\right|g\rangle +|e\rangle )|0\rangle /2$. Here, $|g\rangle$ ($|e\rangle$) and $|n\rangle$ ($n=0,1,2,...$) represent the single-qubit ground (excited) state and cavity photon state in the bare basis. The population undergoes cyclic evolution, and each basis state acquires a different geometric phase. (d) Gate fidelity under the full Hamiltonian equation (1) vs the quality of the approximation conditions ${\mathrm{\Omega }}_i/g_i$ ($i=1,2$). High gate fidelity is achieved when ${\mathrm{\Omega }}_i\gg g_i$ is satisfied.

Figure 3

(a) Schematic pulse sequence for a two-qubit geometric phase gate using $k=5$ pulse segments. (b) The time evolution of the system based on the Hamiltonian in Eq. (2) using the single-segment gate scheme in the transmon parameter regime. The system quantum state is represented in the bare basis indicated in the figure. A large deviation from the ideal case is observed due to the breakdown of the approximations. (c) Simulated gate fidelity based on the Hamiltonian in Eq. (1) for the single-segment (red dots) and multisegment ($k=4$, blue stars) schemes at various bus frequencies and ratios of ${\mathrm{\Omega }}_i/g_i$ (where ${\mathrm{\Omega }}_i$ represents the Rabi frequency in the single-segment scheme). (d) The gate fidelity of the multisegment ($k=4$) scheme at various bus frequencies and qubit frequencies [The first group of bars (left side): Both the qubit frequencies are lower than the bus frequencies. The fifth group of bars (right side): Both qubit frequencies are higher than the bus frequencies. The second to fourth groups of bars: The frequency of $Q_1$ ($Q_2$) is lower (higher) than the bus frequency]. (e) The qubit-state-dependent phase-space trajectories for the bus state using the multisegment ($k=5$) scheme with the higher qubit levels considered. Nodes on each path separate different segments. (f) Gate fidelity vs qubit anharmonicity using the multisegment ($k=5$) scheme with the higher qubit levels considered. The simulation is executed based on the Hamiltonian in Eq. (13). With the further “parameter-search” method to compensate the approximation errors, gate fidelity can always be optimized to above 99.99%. Opt., optimization.

Figure 4

(a) Schematic pulse sequence for an $m$-qubit geometric phase gate ${\mathrm{GP}}_m$ and the generation of an $m$-qubit GHZ state in a single step. The single-qubit rotation $R_x(\pi /2)$ is added for odd $m$. (b) The time evolution of ${\mathrm{GP}}_3$ under the ideal two-level approximation Hamiltonian in Eq. (17) for fluxonium qubits from the initial state $\left(\right|g\rangle +|e\rangle )\left(\right|g\rangle +|e\rangle )\left(\right|g\rangle +|e\rangle )|0\rangle /2\sqrt{2}$. (c) Accumulated three-qubit geometric phases for the multilevel transmon model. Apart from the two-qubit phases, we also obtain a three-body $ZZZ$ term from the qubit-state-dependent force and the frequency shift, which can be useful in quantum simulation of a many-body interaction Hamiltonian. The global phase and the single-qubit terms are not presented here. (d) Gate fidelity vs decoherence time of the system based on the Lindblad master equation [red squares, $T_1$ of the bus where the Lindblad operator is chosen as $a$; orange diamonds, $T_1$ of the qubit where the Lindblad operator is chosen as $a_i$ ($i=1$ or 2); blue stars, $T_2$ of the qubit where the Lindblad operator is chosen as $a_i^{†}{a}_i$ ($i=1$ or 2)]. Here, we assume a total gate time $t_{\mathrm{gate}}=500$ ns. The inset shows an enlarged region of the figure.