Super-robust nonadiabatic geometric quantum control

Nonadiabatic geometric quantum computation (NGQC) and nonadiabatic holonomic quantum computation (NHQC) have been proposed to reduce the run time of geometric quantum gates. However, in terms of robustness against experimental control errors, the existing NGQC and NHQC scenarios have no advantage over standard dynamical gates in most cases. Here, we give the reasons why nonadiabatic geometric gates are sensitive to the control errors and, further, we propose a scheme of super-robust nonadiabatic geometric quantum control, in which the super-robust condition can guarantee both high speed and robustness of the geometric gate. To illustrate the working mechanism of super-robust geometric quantum gates, we give two simple examples of SR-NGQC and SR-NHQC for two- and three-level quantum systems, respectively. Theoretical and numerical results with the experimental parameters indicate that our scheme can significantly improve the gate performance compared to the previous NGQC, NHQC, and standard dynamical schemes. Super-robust geometric quantum computation can be applied to various physical platforms such as superconducting qubits, quantum dots, and trapped ions. All of these sufficiently show that our scheme provides a promising way towards robust geometric quantum computation.

Figure 1

The illustration of our proposed implementation. (a) Conceptual explanation for the ideal nonadiabatic geometric quantum gates in a $(M+N)$-dimensional Hilbert space. The evolution state $|{\mu }_k\left(t\right)\rangle$ acquired a pure nonadiabatic geometric phase ${\gamma }_k$ under a cyclic evolution in computational subspace to construct geometric gates. Meanwhile, the state evolution in a noncomputational basis is completely irrelevant for geometric operations in the Hilbert space. (b) Without super-robust condition (SRC) protection, the fidelity of the nonadiabatic geometric quantum gate with the presence of global control errors is limited because of the unwanted couplings of the time-dependent auxiliary states in the computational and noncomputational subspace. (c) With SRC protection, the effects of the above couplings can be greatly suppressed.

Figure 2

The level structure and coupling configuration for the construction of SR-NHQC and SR-NGQC gates. (a) The driven pulses with amplitudes ${\mathrm{\Omega }}_0$ and ${\mathrm{\Omega }}_1$ resonantly couple $|0\rangle$ and $|1\rangle$ to $|e\rangle$ with the phases ${\varphi }_0$ and ${\varphi }_1$, respectively. (b) The driven pulse with amplitudes ${\mathrm{\Omega }}_R$ resonantly couples $|0\rangle$ and to $|1\rangle$ with the time-independent phase ${\varphi }_R$.

Figure 3

Numerical robustness comparison of the not gate with various approaches. The not gate infidelities $1-F$ of DG, NGQC, SR-NGQC, NHQC, and SR-NHQC are set as a function of the control error of the Rabi frequency, i.e., the relative pulse deviation $\beta$ (a) without and (b) with the decoherence. The not gate fidelities' difference, (d) $\mathrm{\Delta }F=F_G-F_D$ and (c) ($\mathrm{\Delta }F=F_N-F_D$), between the SR-NGQC (NGQC) and DG are set as the decoherence parameter $\mathrm{\Gamma }$ and the relative pulse deviation $\beta$.