Enhanced-Fidelity Ultrafast Geometric Quantum Computation Using Strong Classical Drives

We propose a general approach to implement ultrafast nonadiabatic geometric single- and two-qubit gates by employing counter-rotating effects. This protocol is compatible with most optimal control methods used in previous rotating-wave approximation (RWA) protocols; thus, it is as robust as (or even more robust than) the RWA protocols. Using counter-rotating effects allows us to apply strong drives. Therefore, we can improve the gate speed by 5–10 times compared to the RWA counterpart for implementing high-fidelity ($\ge 99.99\mathrm{\%}$) gates. Such an ultrafast evolution (nanoseconds, even picoseconds) significantly reduces the influence of decoherence (e.g., the qubit dissipation and dephasing). Moreover, because the counter-rotating effects no longer induce a gate infidelity (in both the weak and strong driving regimes), we can achieve a higher fidelity compared to the RWA protocols. Therefore, in the presence of decoherence, one can implement ultrafast geometric quantum gates with $\ge 99\mathrm{\%}$ fidelities.

Figure 1

Population of the ground state $|g⟩$ in the laboratory frame governed by the Hamiltonians under different approximation protocols: $H(t)$ in Eq. (1), $H_{\mathrm{CHRW}}(t)$ in Eq. (12), $H_{\mathrm{RWA}}(t)$ in Eq. (3), and $H_{\text{RWA-BS}}(t)$ in Eq. (4). We choose parameters ${\tilde{\mathrm{\Delta }}}_q={\mathrm{\Delta }}_q=0.1\omega$, ${\mathrm{\Omega }}_0=0.1\omega$, and ${\mathrm{\Omega }}_1=0$. Note that each curve is calculated using the corresponding Hamiltonian after its transformation back to the laboratory frame.

Figure 2

Infidelities $(1-\mathcal{F})$ averaged over time of the CHRW, RWA-BS, and RWA protocols for $T=100/{\mathrm{\Omega }}_0$ with Hamiltonians $H_{\mathrm{CHRW}}(t)$ in Eq. (12), $H_{\mathrm{RWA}}(t)$ in Eq. (3), and $H_{\text{RWA-BS}}(t)$ in Eq. (4), respectively. We choose ${\mathrm{\Omega }}_0(t)=\mathrm{const}$, ${\mathrm{\Delta }}_q(t)={\tilde{\mathrm{\Delta }}}_q(t)=0.1\omega$, and ${\varphi }_0=0$. For the RWA and RWA-BS protocols, we choose ${\mathrm{\Omega }}_1(t)=0$. Hereafter, all the yellow-shaded areas in the figures correspond to high fidelities ($\gtrsim 99.99\mathrm{\%}$); and numerical results are calculated using the Hamiltonian $H(t)$ in Eq. (1).

Figure 3

Implementations of the Hadamard gate using the parameters in Table 1. The parameters shown in (a),(b) are for the CHRW protocol when $T=5\pi /\omega$. The driving amplitude ${\mathrm{\Omega }}_0(t)$ is comparable to the qubit transition frequency ${\omega }_q$ with these parameters.

Figure 4

Comparisons of infidelities ($1-{\overline{F}}_H$) for the three protocols. For a fixed gate time $T$, the RWA protocol has the highest infidelities, while the CHRW protocol has by far the lowest infidelities. For the RWA and RWA-BS protocols, we choose ${\mathrm{\Omega }}_1(t)=0$.

Figure 5

Gate infidelities ($1-{\overline{F}}_H$) of the CHRW, RWA-BS, and RWA protocols for the Hadamard gate. (a) In the presence of dynamical noise [i.e., $\delta {\mathrm{\Delta }}_q(t)=\delta {\mathrm{\Omega }}_0(t)$ are constants]. (b) In the presence of stochastic noise [i.e., $|\delta {\mathrm{\Delta }}_q(t)|,|\delta {\mathrm{\Omega }}_0(t)|\le ϵ$ are random numbers]. We choose the same gate time $T=16\pi /\omega$ for the three protocols, so that the driving amplitudes for different protocols are similar to each others. Other parameters are listed in Table 1.

Figure 6

Gate infidelities: (a) $1-{\overline{F}}_H$ (Hadamard gate) and (b) $1-{\overline{F}}_{\mathrm{CNOT}}$ (cnot-like gate), averaged over 10 000 input states) of the CHRW, RWA-BS, and RWA protocols in the presence of decoherence. We choose the frequency $\omega =2\pi \times 5$ GHz. Other parameters are listed in Tables 1 and 2.

Figure 7

Gate infidelities ($1-\overline{F}$) of the CHRW protocol when the second-excited level (of frequency ${\omega }_2$) of the atom is considered. Parameters for the plot are listed in Tables 1 and 2.

Figure 8

Circuit of a fluxonium qubit, which is formed by a capacitor $C$, an inductor $L$, and a Josephson junction with Josephson energy $E_J$. The qubit is controlled by an external current $I_{\mathrm{out}}(t)$, which couples to the dimensionless flux variable $\mathrm{\Phi }$ through a mutual inductance $M$.