Robustness of controlled Hamiltonian approaches to unitary quantum gates

We examine the effectiveness and resilience of achieving quantum gates employing three approaches stemming from quantum control methods: counterdiabatic driving, Floquet engineering, and inverse engineering. We critically analyze their performance in terms of the gate infidelity, the associated resource overhead based on energetic cost, the susceptibility to timekeeping errors, and the degradation under environmental noise. Despite significant differences in the dynamical path taken, we find a broadly consistent behavior across the three approaches in terms of the efficacy of implementing the target gate and the resource overhead. Furthermore, we establish that the functional form of the control fields plays a crucial role in determining how faithfully a gate operation is achieved. Our results are demonstrated for single-qubit gates, with particular focus on the Hadamard gate, and we discuss the extension to $N$-qubit operations.

Figure 1

(a) The auxiliary control setting based on the protocol from Ref. [51]. Here an auxiliary qubit is coupled to the computational register, with control fields acting only on this auxiliary system. We assume the driven qubit can also experience environmental effects, while the computational qubit is completely isolated. (b) The setup for the inverse engineering setting. The Hamiltonian is designed without any additional resources, and thus the computational qubit is driven directly and can be subject to environmental noise.

Figure 2

We show the trajectories of the qubits for both the CD and IE protocols of the Hadamard gate. We take the initial computational state to be $|+\rangle$. The yellow line corresponds to the path of the qubit in the IE case. The green and orange lines correspond to the computational and auxiliary qubits of the auxiliary evolution cases, respectively, which begin and end with a separable global state of the two qubits, while at intermediate times the reduced states of either qubit are mixed.

Figure 3

(a) Final gate infidelity, given by Eq. (9) as a function of total protocol duration, for the Hadamard gate. The auxiliary evolution with counterdiabatic driving (CD) control is shown in the lowermost green dotted curve. Inverse engineering (IE) performs similarly, shown by the blue dotted curve. The topmost, black dot-dashed curve corresponds to an uncontrolled auxiliary evolution where the performance is several orders of magnitude worse. Floquet engineered (FE) auxiliary control is shown by the red solid curve and is shown to be highly effective. (b) Dynamical gate infidelity for the Hadamard gate with $\tau =1$ using the same styling as (a) to identify the different control protocols. The inset captures the oscillations present in the FE driving around the dynamics of the CD approach. (c) We plot the cost, given by Eq. (10), of implementing the Hadamard gate vs total protocol duration for IE, CD control, FE with $\omega /{\omega }_0=200$ with the same coloring as before. (d) Final gate infidelity for the Hadamard gate vs timekeeping error, $\epsilon$, for over- or undershooting the intended ramp duration. The total (ideal) ramp time is $\tau =1$. We show the performance for the linear (orange), the polynomial (cyan), and the sinusoidal (purple) ramps. In all panels, we fix ${\omega }_0=2\pi /\tau$, $\omega =200{\omega }_0$ for the FE case.

Figure 4

(a) Qubit trajectories for the Hadamard gate under a dephasing channel, given by Eq. (12), where the channel acts on the driven qubit in each case with $\tau \gamma =2$. Styling is the same as in Fig. 2. As we dephase in the $z$ basis, both the state of the qubit in the IE case (yellow) and that of the auxiliary qubit in the CD case (orange) are pulled towards the $z$ axis. The computational qubit of the CD case (green) is not directly affected by the channel and does not deviate from the ideal path, instead stopping along that trajectory once the auxiliary qubit driving the evolution has decohered. (b) We show the final gate infidelity, given by Eq. (9), for the dephasing channel for the Hadamard gate, with upper dotted red and lower dashed blue curves corresponding to the IE and CD cases, respectively.