Dynamically Corrected Nonadiabatic Holonomic Quantum Gates

The key for realizing fault-tolerant quantum computation lies in maintaining the coherence of all qubits so that high-fidelity and robust quantum manipulations on them can be achieved. One of the promising approaches is to use geometric phases in the construction of universal quantum gates, due to their intrinsic robustness against certain types of local noise. However, due to limitations in previous implementations, the noise-resilience feature of nonadiabatic holonomic quantum computation (NHQC) still needs to be improved. Here, in combination with the dynamical-correction technique, we propose a general protocol of universal NHQC with simplified control, which can greatly suppress the effect of the accompanied X errors, retaining the main merit of geometric quantum operations. Numerical simulation shows that the performance of our gate can be much better than that of previous protocols. Remarkably, when incorporating a decoherence-free subspace encoding for the collective dephasing noise, our scheme can also be robust against the involved Z errors. In addition, we also outline the physical implementation of the protocol, which is insensitive to both X and Z errors. Therefore, our protocol provides a promising strategy for scalable fault-tolerant quantum computation.

Figure 1

A schematic illustration of the evolution paths for holonomic quantum gates in Eq. (3) with ${\gamma }_g=\pi /2$. The evolution paths of the bright state $|b⟩$ for the conventional NHQC scheme (a) without and (b) with the systematic X error, which obviously leads to the gate error. The evolution paths of the bright state $|b⟩$ in the DCNHQC scheme (c) without and (d) with the X error. As shown in (d), the X error can be efficiently eliminated by the dynamical-correction process.

Figure 2

The setup of our proposal. (a) A $\mathsf{V}$-type three-level system with resonantly driving $|0⟩\leftrightarrow |a⟩$ and $|1⟩\leftrightarrow |a⟩$ transitions. (b) The coupling configuration for a nontrivial two-qubit holonomic quantum gate. The circles with different colors denote the physical qubits with different frequencies, the squares denote auxiliary physical qubits, and the black solid and orange dashed bonds indicate that the interactions are for single- and two-qubit gates, respectively.

Figure 3

The state population and fidelity dynamics for the dynamically corrected holonomic (a) H and (b) S gates. (c) The shapes of the $\mathrm{\Omega }$ and phases ${\varphi }_0$ in $\mathcal{H}(t)$ for the S gate. (d) The gate robustness with the logarithm to base ten of the gate infidelity for the S gate under the X error without decoherence, obtained from DCNHQC, NHQCOC, conventional single-loop NHQC, and two-loop CNHQC, where the first two techniques show better performance.

Figure 4

The robustness of the S gate in the (a) DCNHQC, (b) conventional single-loop NHQC, (c) two-loop CNHQC, and (d) NHQCOC cases, considering both the X error and the decoherence effect with a uniform decoherence rate $\mathrm{\Gamma }/{\mathrm{\Omega }}_m$. In this realistic case, the proposed DCNHQC strategy performs best.

Figure 5

The gate robustness of the S gate in DCNHQC (a) without and (b) with the DFS encoding, considering both the X error and the collective Z error, where the encoded case performs much better.

Figure 6

The state dynamics of gate $C$ for an initial state of $(|0{⟩}_L+|1{⟩}_L)|0{⟩}_L/\sqrt{2}$.