Superadiabatic holonomic quantum computation in cavity QED

Adiabatic quantum control is a powerful tool for quantum engineering and a key component in some quantum computation models, where accurate control over the timing of the involved pulses is not needed. However, the adiabatic condition requires that the process be very slow and thus limits its application in quantum computation, where quantum gates are preferred to be fast due to the limited coherent times of the quantum systems. Here, we propose a feasible scheme to implement universal holonomic quantum computation based on non-Abelian geometric phases with superadiabatic quantum control, where the adiabatic manipulation is sped up while retaining its robustness against errors in the timing control. Consolidating the advantages of both strategies, our proposal is thus both robust and fast. The cavity QED system is adopted as a typical example to illustrate the merits where the proposed scheme can be realized in a tripod configuration by appropriately controlling the pulse shapes and their relative strength. To demonstrate the distinct performance of our proposal, we also compare our scheme with the conventional adiabatic strategy.

Figure 1

Illustration of the proposed scheme. (a) The level structure and coupling configuration for single-qubit operations, where driven pulses with amplitudes ${\mathrm{\Omega }}_i\left(t\right)$ couple $|i\rangle$ resonantly to $|e\rangle$. (b) Schematic of the level structure in the dressed-state space, where the dark state $|d_2\rangle$ is decoupled from $|+\rangle ,|-\rangle$ and $|d_1\rangle$ when $\phi$ is kept constant. (c) Illustration of a two-qubit gate with two atoms coupled to a cavity in a two-photon Raman resonant way, as shown in (d).

Figure 2

Illustration of the performance of the proposed single gates. The original zero-detuned laser pulses with frequency ${\mathrm{\Omega }}_0\left(t\right)$ (solid black line), ${\mathrm{\Omega }}_1\left(t\right)$ (dashed green line), and ${\mathrm{\Omega }}_2\left(t\right)$ (dotted dashed red line) for the adiabatic single-qubit not and Hadamard gates are sketched in (a) and (b), respectively, with the gate times as $2T$. Qubit-state fidelity dynamics of the not gate (c) and the Hadamard gate (d) as a function of the dimensionless time ${\mathrm{\Omega }}_{max}t/2\pi$. (e) The fidelity of the Hadamard gates for different operation times $T^{{}^{\prime }}$ (in units of 2T). (f) The fidelity of the superadiabatic and modified superadiabatic not gates for different rates of the atom ${\mathrm{\Gamma }}_1^{{}^{\prime }}$ (in units of ${\mathrm{\Gamma }}_1$). $F_N^S$ ($F_N^M$) means the fidelity of not gate when under superadiabatic transitionless driving (under modified superadiabatic driving). $F_H^A,F_H^S,F_H^M$ are the corresponding fidelities of Hadamard gate under adiabatic driving (dotted dashed blue line), superadiabatic transitionless driving (solid green line), and modified superadiabatic driving (dashed red line).

Figure 3

(a) The original pulses for the superadiabatic two-qubit controlled-phase gate. (b) Two-qubit-state fidelity dynamics of the superadiabatic two-qubit cp gate. $F_{CP}^S$ and $F_{CP}^M$ are the corresponding fidelities of controlled-phase gate under superadiabatic transitionless driving and modified superadiabatic driving.

Figure 4

Schematic diagram of the pulses in SATD and MSA schemes for realizing the quantum gate, (a) and (d) for a not gate, (b) and (e) for a Hadamard gate, and two-qubit control phase gate in (c) and (f).