Accelerating adiabatic protocols for entangling two qubits in circuit QED

We introduce a method to speed up adiabatic protocols for creating entanglement between two qubits dispersively coupled to a transmission line, while keeping fidelities high and maintaining robustness to control errors. The method takes genuinely adiabatic sweeps, ranging from a simple Landau-Zener drive to boundary cancellation methods and local adiabatic drivings, and adds fast oscillations to speed up the protocol while canceling unwanted transitions. We compare our protocol with existing adiabatic methods in a state-of-the-art parameter range and show substantial gains. Numerical simulations emphasize that this strategy is efficient also beyond the rotating-wave approximation and that the method is robust against random static biases in the control parameters and with respect to damping and decoherence effects.

Figure 1

Sketch of the general setup. The two qubits, having transition frequencies ${\omega }_1$ and ${\omega }_2$, are coupled with strength $g$ to a resonator of transition frequency ${\omega }_r$, being detuned from it by ${\delta }_1$ and ${\delta }_2$. The two qubits are initially, at time $t=0$, detuned in frequency from each other by ${\omega }_1\left(0\right)-{\omega }_2\left(0\right)=2g{f}_0$ (left). They are then slowly driven into resonance, getting such at time $t=t_f$ (right). Initially, if ${\omega }_1>{\omega }_2$, the (instantaneous) first excited state is $|e\left(0\right)\rangle =|1\downarrow \uparrow \rangle$, while the final (instantaneous) excited state at resonance is $(|1\uparrow \downarrow \rangle +|1\downarrow \uparrow \rangle )/\sqrt{2}$. The instantaneous qubit detuning is described by the sweep function $f\left(t\right)$ which drives the qubits slowly into resonance with each other, so that $f\left(0\right)=f_0$ and $f\left(t_f\right)=0$.

Figure 2

(Rescaled-)time profiles of the adiabatic sweep functions considered in Sec. 2c. Going beyond a simple linear Landau-Zener driving $f_{\text{LZ}}$, the functions $f_{\beta }$ and $f_{\text{PL}}$ represent boundary cancellation methods, while $f_{\text{RC}}$ and $f_{\text{tan}}$ are local adiabatic drivings.

Figure 3

Infidelity [see Eq. (9)] as a function of the total duration of the protocol for different adiabatic sweep functions. The use of boundary cancellation methods $f_{\mathrm{PL}}$ and $f_{\beta }$ (red squares and blue dashed lines, respectively) strongly increases the performance with respect to a simple Landau-Zener sweep $f_{\text{LZ}}$ (black triangles line). For faster times, the local adiabatic drivings $f_{\text{RC}}$ and $f_{\text{tan}}$ (purple circles and green solid lines, respectively) give further improvement.

Figure 4

Imaginary part of the matrix components of $H_{\text{CD}}$ (real parts are zero); see Eq. (11). The (2,3) entry is the dominant one. Taking also into account that this is the nonadiabatic coupling between the two levels realizing the minimal energy gap (in the final configuration), $H_{\text{CD}}^{(2,3)}$ is by far the dominant contributor to nonadiabatic transitions.

Figure 5

Illustration of the construction of the eCD control method. After a reference adiabatic sweep function $f\left(t\right)$ is chosen (a), the corresponding CD Hamiltonian $H_{\text{CD}}$ is computed numerically, having the structure in panel (b). Next, neglecting transitions toward levels not involved in the exploited anticrossing, the corresponding CD corrections are set to zero (c), leading to the Hamiltonian $H_{\text{p}}$. Finally, the latter Hamiltonian is implemented effectively by introducing oscillating control functions in the qubit-resonator couplings (d) that produce the final correcting eCD Hamiltonian $H_{\text{e}}$. The total Hamiltonian for the accelerated adiabatic driving is then $H\left(t\right)+H_{\text{e}}\left(t\right)$.

Figure 6

Comparison between eCD and finite-time adiabatic driving, for a BCM sweep of the form of Eq. (6), and for different ratios $k$ between the maximal amplitude of $H_{\text{e}}$ and the initial qubit-bus coupling $g$; see Eqs. (17) and (18). The squares solid red line shows the results for the unassisted sweep, while the colored (shaded) solid lines correspond to different values of $k$, with an overall (angular) frequency ceiling at $2\pi \times 7$ GHz. The results for different frequency ceilings can be extrapolated using the dashed, dot-dashed, dotted, and dot-dot-dashed curves, which represent the results produced by setting the eCD frequency $\omega /2\pi$ to 4,5,6 and 8 GHz, respectively. The color palette indicates the ratio between the eCD maximal amplitude and $g$.

Figure 7

Comparison between eCD and finite-time adiabatic driving, for a reference LAD of the form of Eq. (8). Different ratios $k$ between the maximal amplitude of $H_{\text{e}}$ and the initial qubit-bus coupling $g$ are considered; see Eqs. (17) and (18). The green solid line shows the results for the unassisted sweep. The colored (shaded) solid curves represent the eCD case for different $k$. The limit imposed by $k$ is further filtered by imposing an overall ceiling in the frequency of the eCD oscillations, set to 7 GHz, for respecting reasonable realizability constraints and consistency with the RWA. The color palette indicates the ratio between eCD maximal amplitude and $g$. The broken black lines represent the infidelities produced by fixing the frequency at 4, 6, and 8 GHz and can be used for extrapolating the results given by different maximal frequency ceilings.

Figure 8

In the top panel, the average relative error on the final infidelity is shown (empty circles), for different total duration of the protocol, when random imperfections within $5\%$ in relative value are simultaneously present in frequency, amplitude, and relative phase [see Eqs. (19) and (20)] of the eCD control functions. The scheme is the LAD + eCD driving described in Sec. 4 with unperturbed eCD frequency fixed at $\omega /2\pi =7$ GHz. In the lower panel, the average logarithmic infidelity for the same data is represented (empty circles), with the solid red line reporting the unperturbed results. In both panels, the error bars extend to one standard deviation, while the asterisk and squared points indicate maximal and minimal values, respectively.

Figure 9

Final fidelity obtained with the proposed control protocol in the presence of damping and decoherence for different values of the decay rates. This is obtained by numerical solution of the master Eq. (21) for a total duration $t_f=100\mathrm{ns}$, assuming equal rates for the two qubits ${\gamma }_r^{\left(1\right)}={\gamma }_r^{\left(2\right)}\equiv \gamma$ and ${\gamma }_{\varphi }^{\left(1\right)}={\gamma }_{\varphi }^{\left(2\right)}=\gamma /2$. For typical experimental values of $\kappa$ and $\gamma$, the fidelity remains around $98\%$, and even letting the rates reach more pessimistic values, it does not drop below $97\%$, proving a good robustness of the control method.