Nonadiabatic geometric quantum gates by composite pulses based on superconducting qubits

The nonadiabatic geometric quantum gate (NGQG) is promising for the realization of high-fidelity operation for large-scale quantum processing. Normally, conventional NGQGs can be especially robust to either the Rabi error or the detuning error, which are two typical errors in many quantum computing platforms. However, it is difficult to suppress these two types of errors at the same time. This remains a big challenge for NGQGs. Here we present a general framework to implement the optimized geometric gate, where the evolution path is performed by using a family of optimized composite pulses. These composite pulses can reduce the sensitivity to the detuning error without introducing an extra Rabi error for this path. Thus, they help fulfill the cyclic evolution condition for the geometric gate. In addition, the inserted composite pulses would not introduce unwanted dynamical phase accumulation. As a result, the designed optimized geometric gate can simultaneously mitigate both the Rabi and detuning errors even in the presence of decoherence related to the real experiments. Our work paves a way to achieve geometric quantum computation robust against multiple errors.

Figure 1

(a) Evolution trajectory of the dressed state $|{\mathrm{\Psi }}_{+}\left(t\right)\rangle$. For the conventional NGQG, it evolves along the path $A\to B\to D\to A$ to enclose the so-called orange-slice shape, while for the optimized geometric gate, it evolves along the path $A\to B\to C\to B\to D\to E\to D\to A$ to fulfill cyclic evolution. (b) For the conventional NGQG, the dressed state from the north pole cannot evolve to the south pole in the presence of detuning error. (c) For the optimized geometric gates, the composite pulses helps the dressed state reach the south pole in the presence of detuning error. (d) Energy level of a transmon qubit, the two lowest levels ($|0\rangle$ and $|1\rangle$) of which are used to encode a single qubit and the third level $|2\rangle$ is regarded as the main leakage source due to the weak anharmonicity in the transmon qubit. (e) Energy-level diagram for two coupled transmon qubits, where the single-excitation subspace $\left\{\right|10\rangle ,|01\rangle \}$ can be used to implement the optimized geometric iswap gate.

Figure 2

Gate fidelity versus detuning error $\delta$, where the Rabi error is zero, i.e., $\epsilon =0$, and the decoherence rate is (a)–(c) $\kappa =0$ and (d)–(f) $\kappa =2\pi \times 4\mathrm{kHz}$, for the (a) and (d) $H$, (b) and (e) $S$, and (c) and (f) $T$ gates. Conventional refers to the geometric gate as shown in Eq. (2b), conventional 2 refers to the scheme as shown in Refs. [46, 47], and optimized refers to the gate parameters in Eq. (2c).

Figure 3

Gate fidelity versus Rabi error $\epsilon$, where the detuning error is zero, i.e., $\delta =0$, and the decoherence rate is (a)–(c) $\kappa =0$ and (d)–(f) $\kappa =2\pi \times 4\mathrm{kHz}$, for the (a) and (d) $H$, (b) and (e) $S$, and (c) and (f) $T$ gates. Conventional refers to the geometric gate as shown in Eq. (2b), conventional 2 refers to the scheme as shown in Refs. [46, 47], and optimized refers to the gate parameters in Eq. (2c).

Figure 4

Infidelity for the effective $\pi$ pulse which corresponds to the evolution operator $U_{BD}(T_2,T_1)$ (blue long-dash–short-dashed lines) in conventional NGQGs and the optimized composite pulses (green dashed lines).

Figure 5

Fidelity of the $H$ gate as a function of decoherence and (a)–(d) detuning error and (e)–(h) Rabi error for (a) and (e) dynamical gates, (b) and (f) the conventional NGQG, (c) and (g) the conventional 2 NGQG, and (d) and (h) the optimized geometric gate.

Figure 6

Physical implementation of the iswap gate in superconducting transmon qubits. (a) The iswap gate as a function of the tunable parameters ${\mathrm{\Delta }}_1$ and ${\beta }_1$. (b) Evolution of the gate fidelity of the optimized iswap gate as a function of time. (c) and (d) Gate fidelity for the detuning and Rabi errors without decoherence effects. Note that all of the above simulation results are based on the full Hamiltonian in Eq. (22).

Figure 7

Fidelity of the iswap gate as a function of the decoherence rate and (a)–(d) Rabi error and (e)–(h) detuning error for the (a) and (e) dynamical gates, (b) and (f) conventional NGQG, (c) and (g) conventional 2 NGQGs, and (d) and (h) optimized geometric gates. Note that all of the above simulation results are based on the Hamiltonian in Eq. (22).